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

<b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:5"/><i><font color="Green">E26</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E2:5"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E3:5"/><i><font color="Green">E28</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:5"/><i><font color="Green">E29</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">pp</font> = 1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font>, <font color="Maroon">qq</font> = 1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E6:5"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">qq</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">pp</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:5"/>
1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/>
<a NAME="E8:5"/><b>then </b>
1 <a href="real_1.html#K5">-</a> <font color="Maroon">pp</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E9:5"/><b>then </b>
<font color="Maroon">q</font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>
<a NAME="E10:5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K2">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a></i>;<br/>
<a NAME="E11:5"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E12:5"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E13:5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T117">XCMPLX_1:117</a></i>;<br/>
<a NAME="E14:5"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">q</font> <a href="real_1.html#K3">+</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E15:5"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E16:5"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<a NAME="E17:5"/><i><font color="Green">E40</font></i>: 
 ex <font color="Olive">g</font> being  <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Olive">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Olive">x</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Olive">g</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Olive">g</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) ) ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="real_1.html#NM1">Real</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:5_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds <br/> ( <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">d</font>] )
 <b>and </b><br/><a NAME="E3:5_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d</font>)
 <b>from </b><i><a class="ref" href="partfun2.html#S2">PARTFUN2:sch 2</a>();<br/></i>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<a NAME="E4:5_1"/><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:5_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E2:5_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:5_1_1"/><b>then </b>
 <a href="numbers.html#K1">REAL</a>  <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:5_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:5_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds  <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<a NAME="E1:5_1_2"/>
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:5_1_2"/>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:5_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Olive">d</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:5_1_3"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>

<b>thus </b><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E7:5_1"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> is  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="fcont_3.html#T4">FCONT_3:4</a>, <a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<a NAME="E8:5_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">g</font> <a href="fdiff_1.html#R2">is_differentiable_on</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a></i>;<br/>
<a NAME="E9:5_1"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Maroon">g</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">g</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>thus </b><a NAME="E1:5_1_4"/>
<font color="Maroon">g</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">d</font>
 <b>by </b><i>, , , <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>

<b>thus </b> <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">g</font>,<font color="Maroon">d</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font>
<b>by </b><i>, , <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> 1
<b>by </b><i>, , , <a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:5_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Olive">x</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Maroon">g</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">g</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) ) ) ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:5"/><i><font color="Green">E52</font></i>: 
 ex <font color="Olive">h</font> being  <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0 &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/>( <font color="Olive">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font> &amp; <font color="Olive">h</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Olive">h</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">h</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span>;<br/>
<a NAME="E1:5_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>take </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span>
;<br/>

<a NAME="E2:5_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 
<b>by </b><i><a class="ref" href="taylor_1.html#D4">TAYLOR_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:5_2"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>

<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:5_2_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:5_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:5_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:5_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, <a class="ref" href="taylor_1.html#D4">TAYLOR_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
;<br/><br/><a NAME="E4:5_2_1"/><i><font color="Green">E59</font></i>: 
 <a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="taylor_1.html#T21">TAYLOR_1:21</a></i>;<br/><b>hence </b><a NAME="E5:5_2_1"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="fdiff_1.html#T23">FDIFF_1:23</a></i>;<br/><b>thus </b> <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">x</font> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T23">FDIFF_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="taylor_1.html#T21">TAYLOR_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>

<br/>.= 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span>

;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:5_2"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font> &amp; <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">g</font> being   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E20:5"/><i><font color="Green">E60</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>and </b><br/><a NAME="E21:5"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Olive">x</font>
 <b>and </b><br/><a NAME="E22:5"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Maroon">g</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">g</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E24:5"/><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>and </b><br/><a NAME="E25:5"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/>( <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font> &amp; <font color="Maroon">h</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">h</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">f</font> = <font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font>;<br/>
<i><font color="Green">E66</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> 0</span>)</span> <a href="subset_1.html#K8">/\</a> <a href="numbers.html#K1">REAL</a> 
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a>, , </i>
<br/>.= 
 <a href="limfunc1.html#K4">right_open_halfline</a> 0
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E27:5"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0 holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Olive">x</font> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:5_4"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 ;<br/>

<a NAME="E2:5_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="real_1.html#NM1">Real</a> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/>
<a NAME="E3:5_4"/><b>then </b>
 ex <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> )
 
;<br/>
<a NAME="E4:5_4"/><b>then </b><i><font color="Green">E70</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font>
 
;<br/>
<a NAME="E5:5_4"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">g</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:5_4"/><i><font color="Green">E72</font></i>: 
 <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">g</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E7:5_4"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">h</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:5_4"/><i><font color="Green">E74</font></i>: 
 <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">h</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font>

;<br/>

<b>thus </b><a NAME="E10:5_4"/>
<font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="holder_1.html#E1:5"><i><font color="Green">E26</font></i></a>, <a class="ref" href="fdiff_1.html#T21">FDIFF_1:21</a></i>;<br/>

<b>thus </b> <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">h</font>,<font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">g</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="txt" href="holder_1.html#E1:5"><i><font color="Green">E26</font></i></a>, <a class="ref" href="fdiff_1.html#T21">FDIFF_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i>, <a class="txt" href="holder_1.html#E2:5"><i><font color="Green">E27</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E28:5"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="prepower.html#T87">PREPOWER:87</a></i>;<br/>
<a NAME="E29:5"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:5_5"/><i><font color="Green">E77</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 ;<br/>

<a NAME="E2:5_5"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 or <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E3:5_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="real_1.html#NM1">Real</a> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:5_5"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 
<b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/>
<a NAME="E5:5_5"/><i><font color="Green">E80</font></i>: 
 <a href="limfunc1.html#K4">right_open_halfline</a> 0 is  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="fcont_3.html#T7">FCONT_3:7</a></i>;<br/>
<a NAME="E6:5_5"/><i><font color="Green">E81</font></i>: 
 <a href="limfunc1.html#K4">right_open_halfline</a> 0 <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:5_5"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0 holds <br/><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:5_5"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font> <a href="fdiff_1.html#R2">is_differentiable_on</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 
<b>by </b><i><a class="txt" href="holder_1.html#E2:5"><i><font color="Green">E27</font></i></a>, <a class="txt" href="holder_1.html#E3:5"><i><font color="Green">E28</font></i></a>, <a class="txt" href="holder_1.html#E4:5"><i><font color="Green">E29</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E9:5_5"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font> <a href="fcont_1.html#R2">is_continuous_on</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T33">FDIFF_1:33</a></i>;<br/>
<a NAME="E10:5_5"/>
1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="real_1.html#NM1">Real</a> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> </span>}</span> 
 
;<br/>
<a NAME="E11:5_5"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 
<b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E85</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="prepower.html#T87">PREPOWER:87</a></i>
<br/>.= 
0
<b>by </b><i/>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_5_2_1"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 or <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_5_2_1_1"><b>case </b></a><a NAME="E1:5_5_2_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">t</font> = 1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font>;<br/><a NAME="E2:5_5_2_1_1"/>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E12:5"><i><font color="Green">E33</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E3:5_5_2_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E4:5_5_2_1_1"/><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:5_5_2_1_1_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><a NAME="E2:5_5_2_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:5_5_2_1_1_1_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> )
 ;<br/><a NAME="E5:5_5_2_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="txt" href="holder_1.html#E14:5"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E6:5_5_2_1_1_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="real_1.html#NM1">Real</a> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="holder_1.html#E14:5"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E7:5_5_2_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:5_5_2_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:5_5_2_1_1"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E6:5_5_2_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:5_5_2_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="holder_1.html#E11:5"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/><a NAME="E8:5_5_2_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="holder_1.html#E6:5"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="fdiff_1.html#T34">FDIFF_1:34</a></i>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:5_5_2_1_1"/><i><font color="Green">E98</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E11:5_5_2_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>and </b><br/><a NAME="E12:5_5_2_1_1"/><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="holder_1.html#T4" target="_self">Th4</a>, <a class="txt" href="holder_1.html#E14:5"><i><font color="Green">E37</font></i></a>, <a class="txt" href="holder_1.html#E15:5"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rolle.html#T4">ROLLE:4</a></i>;<br/><a NAME="E13:5_5_2_1_1"/>
0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="holder_1.html#T4" target="_self">Th4</a>, <a class="txt" href="holder_1.html#E16:5"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E14:5_5_2_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E15:5_5_2_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 ;<br/><a NAME="E16:5_5_2_1_1"/>
<font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="holder_1.html#T4" target="_self">Th4</a>, <a class="txt" href="holder_1.html#E17:5"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E17:5_5_2_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E18:5_5_2_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:5_5_2_1_1_2"/>
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="real_1.html#NM1">Real</a> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:5_5_2_1_1_2"/>
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:5_5_2_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i/>;<br/><a NAME="E20:5_5_2_1_1"/><i><font color="Green">E105</font></i>: 
 <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E21:5_5_2_1_1"/>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E22:5_5_2_1_1"/><b>then </b><i><font color="Green">E106</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E23:5_5_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> 0
 <b>by </b><i>, , , <a class="ref" href="power.html#T45">POWER:45</a></i>;<br/><a NAME="E24:5_5_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> 0
 <b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><a NAME="E25:5_5_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> 0
 <b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><a NAME="E26:5_5_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i>, <a class="ref" href="prepower.html#T85">PREPOWER:85</a></i>;<br/><a NAME="E27:5_5_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E28:5_5_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> 0
 <b>by </b><i><a class="ref" href="holder_1.html#T4" target="_self">Th4</a>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E29:5_5_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E30:5_5_2_1_1"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E31:5_5_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="holder_1.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E32:5_5_2_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="holder_1.html#E1:5"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E33:5_5_2_1_1"/><b>then </b>
0 <a href="real_1.html#K3">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E34:5_5_2_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_5_2_1_2"><b>case </b></a><a NAME="E1:5_5_2_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">t</font> = <font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1;<br/><a NAME="E2:5_5_2_1_2"/>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="holder_1.html#E12:5"><i><font color="Green">E33</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E3:5_5_2_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E4:5_5_2_1_2"/><i><font color="Green">E110</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">1,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5_2_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5_2_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:5_5_2_1_2_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">1,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><a NAME="E2:5_5_2_1_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:5_5_2_1_2_1_1"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span> )
 ;<br/><a NAME="E5:5_5_2_1_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="real_1.html#NM1">Real</a> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="holder_1.html#E14:5"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E6:5_5_2_1_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:5_5_2_1_2_1"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">1,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:5_5_2_1_2"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">1,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">1,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E6:5_5_2_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">1,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:5_5_2_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">1,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="holder_1.html#E11:5"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/><a NAME="E8:5_5_2_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">1,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="holder_1.html#E6:5"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="fdiff_1.html#T34">FDIFF_1:34</a></i>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:5_5_2_1_2"/><i><font color="Green">E116</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E11:5_5_2_1_2"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>and </b><br/><a NAME="E12:5_5_2_1_2"/><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="holder_1.html#T4" target="_self">Th4</a>, <a class="txt" href="holder_1.html#E14:5"><i><font color="Green">E37</font></i></a>, <a class="txt" href="holder_1.html#E15:5"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rolle.html#T4">ROLLE:4</a></i>;<br/><a NAME="E13:5_5_2_1_2"/><i><font color="Green">E118</font></i>: 
0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="holder_1.html#T4" target="_self">Th4</a>, <a class="txt" href="holder_1.html#E16:5"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E14:5_5_2_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E15:5_5_2_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E16:5_5_2_1_2"/><i><font color="Green">E120</font></i>: 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5_2_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:5_5_2_1_2_2"/>
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="real_1.html#NM1">Real</a> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:5_5_2_1_2_2"/>
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:5_5_2_1_2"/>
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E18:5_5_2_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E19:5_5_2_1_2"/><i><font color="Green">E122</font></i>: 
 <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E20:5_5_2_1_2"/>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E21:5_5_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T100">PREPOWER:100</a></i>;<br/><a NAME="E22:5_5_2_1_2"/><b>then </b>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E23:5_5_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="holder_1.html#T4" target="_self">Th4</a>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E24:5_5_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E25:5_5_2_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E17:5"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E26:5_5_2_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="holder_1.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E27:5_5_2_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="holder_1.html#E1:5"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E28:5_5_2_1_2"/><b>then </b>
0 <a href="real_1.html#K3">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E29:5_5_2_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:5_5"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E30:5"/><i><font color="Green">E124</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="prepower.html#T95">PREPOWER:95</a></i>;<br/>
<a NAME="E31:5"/><b>then </b><i><font color="Green">E125</font></i>: 
0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E32:5"/><i><font color="Green">E126</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="prepower.html#T95">PREPOWER:95</a></i>;<br/>
<a NAME="E33:5"/>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_6_1_1"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_6_1_1_1"><b>case </b></a><a NAME="E1:5_6_1_1_1"/>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_6_1_1_1"/>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_6_1_1_2"><b>case </b></a><a NAME="E1:5_6_1_1_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>thus </b><a NAME="E2:5_6_1_1_2"/>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="holder_1.html#E2:5"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:5_6"/>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E34:5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E1:5"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E35:5"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span>
 
;<br/>
<a NAME="E36:5"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="prepower.html#T89">PREPOWER:89</a></i>;<br/>
<a NAME="E37:5"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="prepower.html#T86">PREPOWER:86</a></i>;<br/>
<a NAME="E38:5"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="prepower.html#T92">PREPOWER:92</a></i>;<br/>
<a NAME="E39:5"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="prepower.html#T105">PREPOWER:105</a></i>;<br/>
<a NAME="E40:5"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E41:5"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/>
<a NAME="E42:5"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E43:5"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/>
<a NAME="E44:5"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="prepower.html#T89">PREPOWER:89</a></i>;<br/>
<a NAME="E45:5"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="prepower.html#T85">PREPOWER:85</a></i>;<br/>
<a NAME="E46:5"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>;<br/>
<a NAME="E47:5"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>;<br/>
<a NAME="E48:5"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> iff <font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_7"><b>proof </b></a><div class="add">

<div><i><font color="Green">E130</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_7_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 ;<br/><a NAME="E2:5_7_1"/>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E6:5"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>;<br/><a NAME="E3:5_7_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>;<br/><a NAME="E4:5_7_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T85">PREPOWER:85</a></i>;<br/><a NAME="E5:5_7_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:5_7_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T89">PREPOWER:89</a></i>;<br/><a NAME="E7:5_7_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/><a NAME="E8:5_7_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 ;<br/><a NAME="E9:5_7_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/><a NAME="E10:5_7_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span>
 ;<br/><a NAME="E11:5_7_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T105">PREPOWER:105</a></i>;<br/><a NAME="E12:5_7_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="prepower.html#T92">PREPOWER:92</a></i>;<br/><a NAME="E13:5_7_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T86">PREPOWER:86</a></i>;<br/><a NAME="E14:5_7_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T89">PREPOWER:89</a></i>;<br/><a NAME="E15:5_7_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span>
 ;<br/><a NAME="E16:5_7_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:5"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E17:5_7_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , <a class="txt" href="holder_1.html#E11:5"><i><font color="Green">E31</font></i></a></i>;<br/><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>

<br/>.= 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="holder_1.html#T3" target="_self">Th3</a></i>
;<br/><a NAME="E19:5_7_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T89">PREPOWER:89</a></i>;<br/><a NAME="E20:5_7_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T85">PREPOWER:85</a></i>;<br/><b>hence </b><a NAME="E21:5_7_1"/>
<font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="ref" href="prepower.html#T105">PREPOWER:105</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:5_7"/><i><font color="Green">E134</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="prepower.html#T95">PREPOWER:95</a></i>;<br/>
<a NAME="E3:5_7"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E136</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_7_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_7_2"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:5_7_2"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="prepower.html#T105">PREPOWER:105</a></i>;<br/><font color="Maroon">a</font> = 
<font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> 1
<b>by </b><i>, <a class="ref" href="prepower.html#T86">PREPOWER:86</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E11:5"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, <a class="ref" href="prepower.html#T105">PREPOWER:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E6:5"><i><font color="Green">E30</font></i></a>, <a class="ref" href="holder_1.html#T4" target="_self">Th4</a>, <a class="ref" href="prepower.html#T105">PREPOWER:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> 1
<b>by </b><i><a class="txt" href="holder_1.html#E11:5"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E6:5"><i><font color="Green">E30</font></i></a>, <a class="ref" href="prepower.html#T86">PREPOWER:86</a></i>
;<br/><a NAME="E4:5_7_2"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 ;<br/><a NAME="E5:5_7_2"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T85">PREPOWER:85</a></i>;<br/><a NAME="E6:5_7_2"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T89">PREPOWER:89</a></i>;<br/><a NAME="E7:5_7_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E8:5_7_2"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="holder_1.html#E6:5"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E12:5"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="prepower.html#T89">PREPOWER:89</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="prepower.html#T86">PREPOWER:86</a></i>
;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:5_7"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> iff <font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="txt" href="holder_1.html#E4:5"><i><font color="Green">E29</font></i></a>, <a class="ref" href="holder_1.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E49:5"/>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> &amp; ( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> implies <font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font> ) &amp; ( <font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font> implies <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prepower.html#K13">#R</a> <font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">q</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="holder_1.html#E2:5"><i><font color="Green">E27</font></i></a>, <a class="txt" href="holder_1.html#E3:5"><i><font color="Green">E28</font></i></a></i>;<br/>


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