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

<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>set </b><font color="Maroon" title="c2">Z</font> =  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>) <b>-&gt; </b>   <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>  = $1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1;<br/>
<b>consider </b><font color="Maroon" title="c3">f1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E2:2"/><i><font color="Green" title="E3">A1</font></i>: 
( ( for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] ) ) &amp; ( for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> holds <br/><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">r</font>) ) )
 <b>from </b><i><a class="ref" href="seq_1.html#S3" title="SEQ_1:sch.3">SEQ_1:sch 3</a>();<br/></i>
<a NAME="E3:2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> is   <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> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E4:2"/><b>then </b><i><font color="Green" title="E4">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="entropy1.html#E2:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="fdiff_1.html#T1" title="FDIFF_1:th.1">FDIFF_1:1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <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/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>) <b>-&gt; </b>   <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>  = <span class="p1">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1;<br/>
<b>reconsider </b><font color="Maroon" title="c4">f2</font> =  <a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a>  as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E6:2"/><i><font color="Green" title="E5">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> <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" title="TAYLOR_1:def.2">TAYLOR_1:def 2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <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/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>) <b>-&gt; </b>   <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>  = <span class="p1">(<span class="default">$1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font>;<br/>
<i><font color="Green" title="E6">A4</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0</span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
<b>by </b><i><a class="txt" href="entropy1.html#E6:2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="entropy1.html#E4:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>
<br/>.= 
 <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E8:2"/><i><font color="Green" title="E7">A5</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</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/><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:2_2"/>
<font color="Maroon" title="c6">r</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="E2:2_2"/>
<font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 
<b>by </b><i><a class="txt" href="entropy1.html#E2:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="entropy1.html#E4:2"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:2_2"/>
<font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:2"/><i><font color="Green" title="E8">A6</font></i>: 
( <font color="Maroon" title="c3">f1</font> <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">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</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/> <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c3">f1</font>,<font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3"><b>proof </b></a><div class="add">

<a NAME="E1:2_3"/><i><font color="Green" title="E9">A7</font></i>: 
( <font color="Maroon" title="c3">f1</font> <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">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</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"><font color="Maroon" title="c3">f1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<b>by </b><i><a class="txt" href="entropy1.html#E8:2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="entropy1.html#E4:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="fdiff_1.html#T31" title="FDIFF_1:th.31">FDIFF_1:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:2_3_1"/><i><font color="Green" title="E10">A8</font></i>: 
<font color="Maroon" title="c6">r</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/><b>thus </b> <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c6">r</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">r</font>
<b>by </b><i><a class="txt" href="entropy1.html#E1:2_3"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="entropy1.html#E1:2_3_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="entropy1.html#E1:2_3_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="entropy1.html#E8:2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="entropy1.html#E4:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="fdiff_1.html#T31" title="FDIFF_1:th.31">FDIFF_1:31</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:2_3"/>
( <font color="Maroon" title="c3">f1</font> <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">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</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/> <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c3">f1</font>,<font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <b>by </b><i><a class="txt" href="entropy1.html#E8:2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="entropy1.html#E4:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="fdiff_1.html#T31" title="FDIFF_1:th.31">FDIFF_1:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:2"/><i><font color="Green" title="E9">A9</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font> <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">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</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/> <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Olive" title="b1">r</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:2_4"/><i><font color="Green" title="E10">A10</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font> <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
 <b>by </b><i><a class="txt" href="entropy1.html#E7:2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="fdiff_1.html#T27" title="FDIFF_1:th.27">FDIFF_1:27</a>, <a class="txt" href="entropy1.html#E9:2"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="taylor_1.html#T18" title="TAYLOR_1:th.18">TAYLOR_1:18</a>, <a class="ref" href="taylor_1.html#D3" title="TAYLOR_1:def.3">TAYLOR_1:def 3</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c6">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:2_4_1"/><i><font color="Green" title="E11">A11</font></i>: 
<font color="Maroon" title="c6">r</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/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">r</font> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c6">r</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c6">r</font></span>)</span>
<b>by </b><i><a class="txt" href="entropy1.html#E1:2_4_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="ref" href="fdiff_1.html#T27" title="FDIFF_1:th.27">FDIFF_1:27</a>, <a class="txt" href="entropy1.html#E9:2"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="entropy1.html#E7:2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="taylor_1.html#T18" title="TAYLOR_1:th.18">TAYLOR_1:18</a>, <a class="ref" href="taylor_1.html#D3" title="TAYLOR_1:def.3">TAYLOR_1:def 3</a></i>
<br/>.= 
1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c6">r</font></span>)</span>
<b>by </b><i><a class="txt" href="entropy1.html#E9:2"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="entropy1.html#E1:2_4_1"><i><font color="Green" title="E11">A11</font></i></a></i>
<br/>.= 
1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c6">r</font></span>)</span>
<b>by </b><i><a class="txt" href="entropy1.html#E1:2_4_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="ref" href="taylor_1.html#T18" title="TAYLOR_1:th.18">TAYLOR_1:18</a>, <a class="ref" href="taylor_1.html#D3" title="TAYLOR_1:def.3">TAYLOR_1:def 3</a></i>
;<br/><b>hence </b><a NAME="E3:2_4_1"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c6">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="entropy1.html#E1:2_4"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="entropy1.html#E1:2_4_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E11:2"/>
1 <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> 
 
;<br/>
<a NAME="E12:2"/><b>then </b><i><font color="Green" title="E10">A12</font></i>: 
1 <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#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>;<br/>
<a NAME="E13:2"/><i><font color="Green" title="E11">A13</font></i>: 
(  <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="taylor_1.html#T11" title="TAYLOR_1:th.11">TAYLOR_1:11</a></i>;<br/>
<a NAME="E14:2"/><i><font color="Green" title="E12">A14</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/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:2_5"/><i><font color="Green" title="E13">A15</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<a NAME="E2:2_5"/>
<font color="Maroon" title="c6">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="entropy1.html#E1:2_5"><i><font color="Green" title="E13">A15</font></i></a></i>;<br/>
<a NAME="E3:2_5"/><b>then </b><i><font color="Green" title="E14">A16</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="entropy1.html#E7:2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="limfunc1.html#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>;<br/>
<a NAME="E4:2_5"/><b>then </b><i><font color="Green" title="E15">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E5:2_5"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="entropy1.html#E3:2_5"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E6:2_5"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:2_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="entropy1.html#E2:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="entropy1.html#E4:2_5"><i><font color="Green" title="E15">A17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E13">A18</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 = 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1</span>)</span>

<br/>.= 
 <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1</span>)</span>

<br/>.= 
 <a href="binop_2.html#K7" title="BINOP_2:func.7">-</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> ,1</span>)</span>
<b>by </b><i><a class="txt" href="entropy1.html#E12:2"><i><font color="Green" title="E10">A12</font></i></a>, <a class="ref" href="taylor_1.html#D2" title="TAYLOR_1:def.2">TAYLOR_1:def 2</a></i>
<br/>.= 
 <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 0
<b>by </b><i><a class="txt" href="entropy1.html#E13:2"><i><font color="Green" title="E11">A13</font></i></a>, <a class="ref" href="power.html#T59" title="POWER:th.59">POWER:59</a></i>
<br/>.= 
0

;<br/>
<b>set </b><font color="Maroon" title="c6">Z1</font> =  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1;<br/>
<a NAME="E16:2"/><i><font color="Green" title="E14">A19</font></i>: 
 <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#T8" title="LIMFUNC1:th.8">LIMFUNC1:8</a></i>;<br/>
<a NAME="E17:2"/><b>then </b><i><font color="Green" title="E15">A20</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font> <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> 1
 
<b>by </b><i><a class="txt" href="entropy1.html#E10:2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</a></i>;<br/>
<a NAME="E18:2"/><i><font color="Green" title="E16">A21</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</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> 1 holds <br/> <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:2_7"/><i><font color="Green" title="E17">A22</font></i>: 
<font color="Maroon" title="c7">r</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> 1
 ;<br/>

<a NAME="E2:2_7"/>
<font color="Maroon" title="c7">r</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> : 1 <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#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a>, <a class="txt" href="entropy1.html#E1:2_7"><i><font color="Green" title="E17">A22</font></i></a></i>;<br/>
<a NAME="E3:2_7"/><b>then </b>
 ex <font color="Olive" title="b1">g1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g1</font> )
 
;<br/>
<a NAME="E4:2_7"/><b>then </b><i><font color="Green" title="E18">A23</font></i>: 
1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="square_1.html#T2" title="SQUARE_1:th.2">SQUARE_1:2</a></i>;<br/>
<a NAME="E5:2_7"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c7">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="entropy1.html#E16:2"><i><font color="Green" title="E14">A19</font></i></a>, <a class="txt" href="entropy1.html#E1:2_7"><i><font color="Green" title="E17">A22</font></i></a>, <a class="txt" href="entropy1.html#E10:2"><i><font color="Green" title="E9">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:2_7"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E4:2_7"><i><font color="Green" title="E18">A23</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c7">Z2</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/>
<a NAME="E19:2"/><i><font color="Green" title="E17">A24</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#T11" title="LIMFUNC1:th.11">LIMFUNC1:11</a></i>;<br/>
<a NAME="E20:2"/><b>then </b><i><font color="Green" title="E18">A25</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="entropy1.html#E10:2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</a></i>;<br/>
<a NAME="E21:2"/><i><font color="Green" title="E19">A26</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:2_8"/><i><font color="Green" title="E20">A27</font></i>: 
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/>

<a NAME="E2:2_8"/><b>then </b>
 ex <font color="Olive" title="b1">g1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 
;<br/>
<a NAME="E3:2_8"/><b>then </b><i><font color="Green" title="E21">A28</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c8">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T189" title="XREAL_1:th.189">XREAL_1:189</a></i>;<br/>
<a NAME="E4:2_8"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c8">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="entropy1.html#E19:2"><i><font color="Green" title="E17">A24</font></i></a>, <a class="txt" href="entropy1.html#E1:2_8"><i><font color="Green" title="E20">A27</font></i></a>, <a class="txt" href="entropy1.html#E10:2"><i><font color="Green" title="E9">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:2_8"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E3:2_8"><i><font color="Green" title="E21">A28</font></i></a>, <a class="ref" href="xreal_1.html#T51" title="XREAL_1:th.51">XREAL_1:51</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:2"/><i><font color="Green" title="E20">A29</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0 &amp; <font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 &amp; <font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 
<b>by </b><i><a class="txt" href="entropy1.html#E10:2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="entropy1.html#E17:2"><i><font color="Green" title="E15">A20</font></i></a>, <a class="txt" href="entropy1.html#E20:2"><i><font color="Green" title="E18">A25</font></i></a>, <a class="ref" href="fdiff_1.html#T33" title="FDIFF_1:th.33">FDIFF_1:33</a></i>;<br/>
<a NAME="E23:2"/><i><font color="Green" title="E21">A30</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</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> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_9"/>
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</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> 1 &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:2_9"/><i><font color="Green" title="E22">A31</font></i>: 
( <font color="Maroon" title="c8">r1</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> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 )
 ;<br/>
<a NAME="E4:2_9"/>
<font color="Maroon" title="c8">r1</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> : 1 <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#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a>, <a class="txt" href="entropy1.html#E3:2_9"><i><font color="Green" title="E22">A31</font></i></a></i>;<br/>
<a NAME="E5:2_9"/><b>then </b><i><font color="Green" title="E23">A32</font></i>: 
 ex <font color="Olive" title="b1">g1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g1</font> )
 
;<br/>
<a NAME="E6:2_9"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">1,<font color="Maroon" title="c8">r1</font></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="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0
 
<b>by </b><i><a class="ref" href="limfunc1.html#T21" title="LIMFUNC1:th.21">LIMFUNC1:21</a></i>;<br/>
<a NAME="E7:2_9"/><b>then </b><i><font color="Green" title="E24">A33</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">1,<font color="Maroon" title="c8">r1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="entropy1.html#E22:2"><i><font color="Green" title="E20">A29</font></i></a>, <a class="ref" href="fcont_1.html#T17" title="FCONT_1:th.17">FCONT_1:17</a></i>;<br/>
<a NAME="E8:2_9"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">1,<font color="Maroon" title="c8">r1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#T20" title="LIMFUNC1:th.20">LIMFUNC1:20</a></i>;<br/>
<a NAME="E9:2_9"/><b>then </b><i><font color="Green" title="E25">A34</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">1,<font color="Maroon" title="c8">r1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="entropy1.html#E10:2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_9_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="entropy1.html#E3:2_9"><i><font color="Green" title="E22">A31</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_9_1_1"><b>suppose </b></a><a NAME="E1:2_9_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:2_9_1_1"/><i><font color="Green" title="E26">A35</font></i>: 
( <font color="Maroon" title="c9">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">1,<font color="Maroon" title="c8">r1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c9">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="entropy1.html#E15:2"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="entropy1.html#E5:2_9"><i><font color="Green" title="E23">A32</font></i></a>, <a class="txt" href="entropy1.html#E7:2_9"><i><font color="Green" title="E24">A33</font></i></a>, <a class="txt" href="entropy1.html#E9:2_9"><i><font color="Green" title="E25">A34</font></i></a>, <a class="ref" href="rolle.html#T1" title="ROLLE:th.1">ROLLE:1</a></i>;<br/><a NAME="E4:2_9_1_1"/>
 ex <font color="Olive" title="b1">g1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">r1</font> )
 <b>by </b><i><a class="txt" href="entropy1.html#E3:2_9_1_1"><i><font color="Green" title="E26">A35</font></i></a></i>;<br/><a NAME="E5:2_9_1_1"/><b>then </b>
<font color="Maroon" title="c9">r2</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> : 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g</font> </span>}</span> 
 ;<br/><a NAME="E6:2_9_1_1"/><b>then </b>
<font color="Maroon" title="c9">r2</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> 1
 <b>by </b><i><a class="ref" href="limfunc1.html#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>;<br/><b>hence </b><a NAME="E7:2_9_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="entropy1.html#E3:2_9_1_1"><i><font color="Green" title="E26">A35</font></i></a>, <a class="txt" href="entropy1.html#E18:2"><i><font color="Green" title="E16">A21</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_9_1_2"><b>suppose </b></a><a NAME="E1:2_9_1_2"/><i><font color="Green" title="E26">A36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c9">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:2_9_1_2"/><i><font color="Green" title="E27">A37</font></i>: 
( <font color="Maroon" title="c9">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">1,<font color="Maroon" title="c8">r1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c9">r3</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"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="entropy1.html#E5:2_9"><i><font color="Green" title="E23">A32</font></i></a>, <a class="txt" href="entropy1.html#E7:2_9"><i><font color="Green" title="E24">A33</font></i></a>, <a class="txt" href="entropy1.html#E9:2_9"><i><font color="Green" title="E25">A34</font></i></a>, <a class="ref" href="rolle.html#T3" title="ROLLE:th.3">ROLLE:3</a></i>;<br/><a NAME="E4:2_9_1_2"/>
 ex <font color="Olive" title="b1">g1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r3</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">r1</font> )
 <b>by </b><i><a class="txt" href="entropy1.html#E3:2_9_1_2"><i><font color="Green" title="E27">A37</font></i></a></i>;<br/><a NAME="E5:2_9_1_2"/><b>then </b>
<font color="Maroon" title="c9">r3</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> : 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g</font> </span>}</span> 
 ;<br/><a NAME="E6:2_9_1_2"/><b>then </b><i><font color="Green" title="E28">A38</font></i>: 
<font color="Maroon" title="c9">r3</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> 1
 <b>by </b><i><a class="ref" href="limfunc1.html#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>;<br/><a NAME="E7:2_9_1_2"/>
<font color="Maroon" title="c8">r1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E5:2_9"><i><font color="Green" title="E23">A32</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E8:2_9_1_2"/><b>then </b>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c9">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E3:2_9_1_2"><i><font color="Green" title="E27">A37</font></i></a>, <a class="txt" href="entropy1.html#E1:2_9_1_2"><i><font color="Green" title="E26">A36</font></i></a>, <a class="txt" href="entropy1.html#E15:2"><i><font color="Green" title="E13">A18</font></i></a>, <a class="ref" href="real_2.html#T128" title="REAL_2:th.128">REAL_2:128</a></i>;<br/><b>hence </b><a NAME="E9:2_9_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="entropy1.html#E6:2_9_1_2"><i><font color="Green" title="E28">A38</font></i></a>, <a class="txt" href="entropy1.html#E18:2"><i><font color="Green" title="E16">A21</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E24:2"/><i><font color="Green" title="E22">A39</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_10"/>
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:2_10"/><i><font color="Green" title="E23">A40</font></i>: 
( <font color="Maroon" title="c8">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 )
 ;<br/>
<a NAME="E4:2_10"/><i><font color="Green" title="E24">A41</font></i>: 
 ex <font color="Olive" title="b1">g1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r1</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 
<b>by </b><i><a class="txt" href="entropy1.html#E3:2_10"><i><font color="Green" title="E23">A40</font></i></a></i>;<br/>
<a NAME="E5:2_10"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c8">r1</font>,1</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="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0
 
<b>by </b><i><a class="ref" href="limfunc1.html#T21" title="LIMFUNC1:th.21">LIMFUNC1:21</a></i>;<br/>
<a NAME="E6:2_10"/><b>then </b><i><font color="Green" title="E25">A42</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c8">r1</font>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="entropy1.html#E22:2"><i><font color="Green" title="E20">A29</font></i></a>, <a class="ref" href="fcont_1.html#T17" title="FCONT_1:th.17">FCONT_1:17</a></i>;<br/>
<a NAME="E7:2_10"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c8">r1</font>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0
 
<b>by </b><i><a class="txt" href="entropy1.html#E4:2_10"><i><font color="Green" title="E24">A41</font></i></a>, <a class="ref" href="limfunc1.html#T20" title="LIMFUNC1:th.20">LIMFUNC1:20</a></i>;<br/>
<a NAME="E8:2_10"/><b>then </b><i><font color="Green" title="E26">A43</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c8">r1</font>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="entropy1.html#E10:2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_10_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="entropy1.html#E3:2_10"><i><font color="Green" title="E23">A40</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_10_1_1"><b>suppose </b></a><a NAME="E1:2_10_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:2_10_1_1"/><i><font color="Green" title="E27">A44</font></i>: 
( <font color="Maroon" title="c9">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c8">r1</font>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c9">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="entropy1.html#E15:2"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="entropy1.html#E4:2_10"><i><font color="Green" title="E24">A41</font></i></a>, <a class="txt" href="entropy1.html#E6:2_10"><i><font color="Green" title="E25">A42</font></i></a>, <a class="txt" href="entropy1.html#E8:2_10"><i><font color="Green" title="E26">A43</font></i></a>, <a class="ref" href="rolle.html#T1" title="ROLLE:th.1">ROLLE:1</a></i>;<br/><a NAME="E4:2_10_1_1"/>
 ex <font color="Olive" title="b1">g1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font> &amp; <font color="Maroon" title="c8">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="entropy1.html#E3:2_10_1_1"><i><font color="Green" title="E27">A44</font></i></a></i>;<br/><a NAME="E5:2_10_1_1"/><b>then </b>
<font color="Maroon" title="c9">r2</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> &amp; <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="entropy1.html#E4:2_10"><i><font color="Green" title="E24">A41</font></i></a></i>;<br/><b>hence </b><a NAME="E6:2_10_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="entropy1.html#E3:2_10_1_1"><i><font color="Green" title="E27">A44</font></i></a>, <a class="txt" href="entropy1.html#E21:2"><i><font color="Green" title="E19">A26</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_10_1_2"><b>suppose </b></a><a NAME="E1:2_10_1_2"/><i><font color="Green" title="E27">A45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c9">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:2_10_1_2"/><i><font color="Green" title="E28">A46</font></i>: 
( <font color="Maroon" title="c9">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c8">r1</font>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c9">r3</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"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r1</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c8">r1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="entropy1.html#E4:2_10"><i><font color="Green" title="E24">A41</font></i></a>, <a class="txt" href="entropy1.html#E6:2_10"><i><font color="Green" title="E25">A42</font></i></a>, <a class="txt" href="entropy1.html#E8:2_10"><i><font color="Green" title="E26">A43</font></i></a>, <a class="ref" href="rolle.html#T3" title="ROLLE:th.3">ROLLE:3</a></i>;<br/><a NAME="E4:2_10_1_2"/>
 ex <font color="Olive" title="b1">g1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r3</font> &amp; <font color="Maroon" title="c8">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="entropy1.html#E3:2_10_1_2"><i><font color="Green" title="E28">A46</font></i></a></i>;<br/><a NAME="E5:2_10_1_2"/><b>then </b><i><font color="Green" title="E29">A47</font></i>: 
<font color="Maroon" title="c9">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="entropy1.html#E4:2_10"><i><font color="Green" title="E24">A41</font></i></a></i>;<br/><a NAME="E6:2_10_1_2"/><i><font color="Green" title="E30">A48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E1:2_10_1_2"><i><font color="Green" title="E27">A45</font></i></a>, <a class="txt" href="entropy1.html#E15:2"><i><font color="Green" title="E13">A18</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E7:2_10_1_2"/>
1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c8">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E4:2_10"><i><font color="Green" title="E24">A41</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E8:2_10_1_2"/><b>then </b>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c9">r3</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E3:2_10_1_2"><i><font color="Green" title="E28">A46</font></i></a>, <a class="txt" href="entropy1.html#E6:2_10_1_2"><i><font color="Green" title="E30">A48</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/><b>hence </b><a NAME="E9:2_10_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="entropy1.html#E5:2_10_1_2"><i><font color="Green" title="E29">A47</font></i></a>, <a class="txt" href="entropy1.html#E21:2"><i><font color="Green" title="E19">A26</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E25:2"/><i><font color="Green" title="E23">A49</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:2_11"/><i><font color="Green" title="E24">A50</font></i>: 
( <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 ;<br/>

<a NAME="E2:2_11"/>
<font color="Maroon" title="c8">r</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> &amp; <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="entropy1.html#E1:2_11"><i><font color="Green" title="E24">A50</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:2_11"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E24:2"><i><font color="Green" title="E22">A39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:2"/><i><font color="Green" title="E24">A51</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:2_12"/><i><font color="Green" title="E25">A52</font></i>: 
<font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/>

<a NAME="E2:2_12"/>
<font color="Maroon" title="c8">r</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> : 1 <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="entropy1.html#E1:2_12"><i><font color="Green" title="E25">A52</font></i></a></i>;<br/>
<a NAME="E3:2_12"/><b>then </b>
<font color="Maroon" title="c8">r</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> 1
 
<b>by </b><i><a class="ref" href="limfunc1.html#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>;<br/>
<b>hence </b><a NAME="E4:2_12"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E23:2"><i><font color="Green" title="E21">A30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:2"/><i><font color="Green" title="E25">A53</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:2_13"/><i><font color="Green" title="E26">A54</font></i>: 
<font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_13_1"/>
( <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 or <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_13_1_1"><b>suppose </b></a><a NAME="E1:2_13_1_1"/>
<font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_13_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E1:2_13"><i><font color="Green" title="E26">A54</font></i></a>, <a class="txt" href="entropy1.html#E25:2"><i><font color="Green" title="E23">A49</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_13_1_2"><b>suppose </b></a><a NAME="E1:2_13_1_2"/>
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_13_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E15:2"><i><font color="Green" title="E13">A18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_13_1_3"><b>suppose </b></a><a NAME="E1:2_13_1_3"/>
<font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_13_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E26:2"><i><font color="Green" title="E24">A51</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>assume </b><a NAME="E28:2"/><i><font color="Green" title="E26">A55</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<a NAME="E29:2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="entropy1.html#E27:2"><i><font color="Green" title="E25">A53</font></i></a></i>;<br/>
<a NAME="E30:2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="entropy1.html#E28:2"><i><font color="Green" title="E26">A55</font></i></a>, <a class="txt" href="entropy1.html#E14:2"><i><font color="Green" title="E12">A14</font></i></a></i>;<br/>
<a NAME="E31:2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 0 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></i>;<br/>
<b>hence </b><a NAME="E32:2"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 ;<br/>

<b>thus </b><a NAME="E33:2"/><i><font color="Green" title="E27">A56</font></i>: 
( <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <span class="p1">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_14"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:2_14_1"/>
<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:2_14_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E15:2"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="entropy1.html#E14:2"><i><font color="Green" title="E12">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E3:2_14_1"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:2_14"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 ;<br/>

<a NAME="E3:2_14"/><b>then </b><i><font color="Green" title="E28">A57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
;<br/>
<b>assume </b><a NAME="E4:2_14"/><i><font color="Green" title="E29">A58</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_14_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_14_2"/>
( <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 or <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="entropy1.html#E4:2_14"><i><font color="Green" title="E29">A58</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_14_2_1"><b>suppose </b></a><a NAME="E1:2_14_2_1"/>
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_14_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E25:2"><i><font color="Green" title="E23">A49</font></i></a>, <a class="txt" href="entropy1.html#E28:2"><i><font color="Green" title="E26">A55</font></i></a></i>;<br/><b>hence </b><a NAME="E3:2_14_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="entropy1.html#E28:2"><i><font color="Green" title="E26">A55</font></i></a>, <a class="txt" href="entropy1.html#E14:2"><i><font color="Green" title="E12">A14</font></i></a>, <a class="txt" href="entropy1.html#E3:2_14"><i><font color="Green" title="E28">A57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_14_2_2"><b>suppose </b></a><a NAME="E1:2_14_2_2"/><i><font color="Green" title="E30">A59</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_14_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E26:2"><i><font color="Green" title="E24">A51</font></i></a></i>;<br/><b>hence </b><a NAME="E3:2_14_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="entropy1.html#E3:2_14"><i><font color="Green" title="E28">A57</font></i></a>, <a class="txt" href="entropy1.html#E1:2_14_2_2"><i><font color="Green" title="E30">A59</font></i></a>, <a class="txt" href="entropy1.html#E14:2"><i><font color="Green" title="E12">A14</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E34:2"/>
( <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 iff <span class="p1">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_15"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:2_15_1"/>
<font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:2_15_1"/><b>then </b><i><font color="Green" title="E28">A60</font></i>: 
( <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T57" title="XREAL_1:th.57">XREAL_1:57</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_15_1_1"/>
( <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 or <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="entropy1.html#E2:2_15_1"><i><font color="Green" title="E28">A60</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_15_1_1_1"><b>suppose </b></a><a NAME="E1:2_15_1_1_1"/>
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_15_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E25:2"><i><font color="Green" title="E23">A49</font></i></a>, <a class="txt" href="entropy1.html#E28:2"><i><font color="Green" title="E26">A55</font></i></a></i>;<br/><a NAME="E3:2_15_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E28:2"><i><font color="Green" title="E26">A55</font></i></a>, <a class="txt" href="entropy1.html#E14:2"><i><font color="Green" title="E12">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E4:2_15_1_1_1"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_15_1_1_2"><b>suppose </b></a><a NAME="E1:2_15_1_1_2"/><i><font color="Green" title="E29">A61</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_15_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E26:2"><i><font color="Green" title="E24">A51</font></i></a></i>;<br/><a NAME="E3:2_15_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E1:2_15_1_1_2"><i><font color="Green" title="E29">A61</font></i></a>, <a class="txt" href="entropy1.html#E14:2"><i><font color="Green" title="E12">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E4:2_15_1_1_2"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:2_15"/><i><font color="Green" title="E28">A62</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K2" title="TAYLOR_1:func.2">log_</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 ;<br/>

<b>assume </b><a NAME="E3:2_15"/>
<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>hence </b><a NAME="E4:2_15"/>
contradiction
 <b>by </b><i><a class="txt" href="entropy1.html#E2:2_15"><i><font color="Green" title="E28">A62</font></i></a>, <a class="txt" href="entropy1.html#E33:2"><i><font color="Green" title="E27">A56</font></i></a></i>;<br/>


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


</div>
