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

<b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:50"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K31">compreal</a> 
 ;<br/>

<b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<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>  =  <a href="binop_2.html#K7">-</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">f1</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="E3:50"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] )
 <b>and </b><br/><a NAME="E4:50"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> holds <br/><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">p</font>)
 <b>from </b><i><a class="ref" href="partfun2.html#S2">PARTFUN2:sch 2</a>();<br/></i>
<a NAME="E5:50"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">f1</font> is   <a href="fdiff_1.html#NM2">LINEAR</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_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="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="txt" href="sin_cos2.html#E4"><i><font color="Green">Lemma30</font></i></a></i>;<br/>
<a NAME="E2:50_1_1"/><b>then </b>
 <a href="numbers.html#K1">REAL</a>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:50_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</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="E2:50_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">f1</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E3:50_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds  <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2"><b>proof </b></a><div class="add">

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

<a NAME="E1:50_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="txt" href="sin_cos2.html#E4"><i><font color="Green">Lemma30</font></i></a></i>;<br/>
<a NAME="E2:50_1_2"/><b>then </b>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:50_1_2"/>
<font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">p</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="E4:50_1"/>
for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_3"><b>proof </b></a><div class="add">

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

<a NAME="E1:50_1_3"/>
<font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:50_1_3"/>
<font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">p</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:50_1"/>
<font color="Maroon">f1</font> is   <a href="fdiff_1.html#NM2">LINEAR</a>
 <b>by </b><i><a class="ref" href="sin_cos2.html#D1" target="_self">Def1</a>, <a class="ref" href="fdiff_1.html#D4">FDIFF_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">f2</font> = <a href="numbers.html#K1">REAL</a>  <a href="funcop_1.html#K2">--&gt;</a> 0 as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E7:50"/><i><font color="Green">E68</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E8:50"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds  <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E9:50"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">f2</font> is   <a href="fdiff_1.html#NM1">REST</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<a NAME="E1:50_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a>, <a class="ref" href="sin_cos2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:50_2"/>
<font color="Maroon">f2</font> is  <a href="fdiff_1.html#V2">REST-like</a> <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_2_1"/>
for <font color="Olive">hy1</font> being <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Olive">hy1</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Olive">hy1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">hy1</font> be  <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<a NAME="E1:50_2_1_1"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<i><font color="Green">E74</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
;<br/>
<a NAME="E2:50_2_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">hy1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 
<b>by </b><i><a class="ref" href="sin_cos2.html#D1" target="_self">Def1</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> 0
<b>by </b><i/>
<br/>.= 
0

;<br/>
<b>hence </b><a NAME="E4:50_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:50_2_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D6">SEQM_3:def 6</a></i>;<br/>
<a NAME="E3:50_2_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="seq_4.html#T41">SEQ_4:41</a></i>
<br/>.= 
0
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:50_2_1_1_2"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:50_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hy1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">hy1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:50_2_1"/>
<font color="Maroon">f2</font> is  <a href="fdiff_1.html#V2">REST-like</a> <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, <a class="ref" href="fdiff_1.html#D3">FDIFF_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:50_2"/>
<font color="Maroon">f2</font> is   <a href="fdiff_1.html#NM1">REST</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:50"/><i><font color="Green">E76</font></i>: 
 ex <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">p</font> st <br/>( <font color="Olive">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="binop_2.html#K31">compreal</a>  &amp; ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> holds <br/><span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">r</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><b>proof </b></a><div class="add">

<a NAME="E1:50_3"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">r</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<a NAME="E1:50_3_1"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<a NAME="E1:50_3_1_1"/>
<font color="Maroon">p</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E2:50_3_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="txt" href="sin_cos2.html#E4"><i><font color="Green">Lemma30</font></i></a></i>;<br/>
<a NAME="E3:50_3_1_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:50_3_1_1"/>
<font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</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="E2:50_3_1"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">d</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<a NAME="E1:50_3_1_2"/>
<font color="Maroon">d</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E2:50_3_1_2"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 
<b>by </b><i><a class="ref" href="sin_cos2.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E3:50_3_1_2"/>
<font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos2.html#E3:50"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i><a class="txt" href="sin_cos2.html#E3:50"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E4:50_3_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> implies <span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>take </b>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
;<br/>

<b>thus </b><a NAME="E2:50_3"/>
( <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> is    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">p</font> &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="binop_2.html#K31">compreal</a>  &amp; ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="seq_1.html#K2">.</a> <font color="Olive">r</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="txt" href="sin_cos2.html#E4:50"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:50"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">f</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="sin_cos2.html#T1" target="_self">Th1</a>, , , <a class="ref" href="fdiff_1.html#D5">FDIFF_1:def 5</a></i>;<br/>
<a NAME="E12:50"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds  <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4"><b>proof </b></a><div class="add">

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

<a NAME="E1:50_4"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="txt" href="sin_cos2.html#E4"><i><font color="Green">Lemma30</font></i></a></i>;<br/>
<a NAME="E2:50_4"/><b>then </b>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:50_4"/>
<font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">p</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 href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">p</font> = 
<font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> 1
<b>by </b><i><a class="ref" href="sin_cos2.html#T1" target="_self">Th1</a>, , , , , <a class="ref" href="fdiff_1.html#D6">FDIFF_1:def 6</a></i>
<br/>.= 
 <a href="binop_2.html#K7">-</a> 1
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E14:50"/>
( <font color="Maroon">f</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">p</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 <b>by </b><i><a class="ref" href="sin_cos2.html#T1" target="_self">Th1</a>, , , , <a class="ref" href="fdiff_1.html#D5">FDIFF_1:def 5</a></i>;<br/>


</div>
