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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:51"/><i><font color="Green">E44</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E3:51"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1"><b>case </b></a><a NAME="E1:51_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:51_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/><a NAME="E3:51_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:51"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/><a NAME="E4:51_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E3:51"><i><font color="Green">E46</font></i></a>, <a class="ref" href="rcomp_1.html#T17">RCOMP_1:17</a></i>;<br/><a NAME="E5:51_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></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="E6:51_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E4:51_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:51_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E8:51_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:51"><i><font color="Green">E45</font></i></a>, <a class="ref" href="taylor_1.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E9:51_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E10:51_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:51"><i><font color="Green">E45</font></i></a>, <a class="ref" href="taylor_1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E11:51_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="fdiff_1.html#T33">FDIFF_1:33</a></i>;<br/><a NAME="E12:51_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>1</sub></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E4:51_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/><a NAME="E13:51_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:51"><i><font color="Green">E45</font></i></a>, <a class="txt" href="taylor_1.html#E6:51_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="taylor_1.html#T31" target="_self">Th31</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E15:51_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>and </b><br/><a NAME="E16:51_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</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"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:51_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="taylor_1.html#E4:51_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="taylor_1.html#E8:51_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="taylor_1.html#E12:51_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="taylor_1.html#T29" target="_self">Th29</a></i>;<br/><i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:51"><i><font color="Green">E45</font></i></a>, <a class="txt" href="taylor_1.html#E6:51_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="taylor_1.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E15:51_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E18:51_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:51_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/><a NAME="E19:51_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="taylor_1.html#E15:51_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><a NAME="E20:51_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E21:51_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E22:51_1_1_1"/><b>then </b>
0 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E18:51_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E23:51_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E24:51_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E20:51_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E25:51_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E26:51_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E18:51_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E27:51_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i><a class="txt" href="taylor_1.html#E18:51_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E18:51_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>

;<br/><b>hence </b><a NAME="E29:51_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</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"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E16:51_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="taylor_1.html#E17:51_1_1_1"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_2"><b>case </b></a><a NAME="E1:51_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>6</sub></font> = 1 <a href="real_1.html#K6">/</a> 2;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = 1 <a href="real_1.html#K6">/</a> 2;<br/><b>thus </b><a NAME="E2:51_1_1_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 ;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</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"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</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"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:51"><i><font color="Green">E46</font></i></a>, <a class="txt" href="taylor_1.html#E1:51_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="taylor_1.html#T32" target="_self">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</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"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">0 <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>

;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_3"><b>case </b></a><a NAME="E1:51_1_1_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:51_1_1_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/><a NAME="E3:51_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:51"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/><a NAME="E4:51_1_1_3"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E3:51"><i><font color="Green">E46</font></i></a>, <a class="ref" href="rcomp_1.html#T17">RCOMP_1:17</a></i>;<br/><a NAME="E5:51_1_1_3"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></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="E6:51_1_1_3"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E4:51_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:51_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E8:51_1_1_3"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:51"><i><font color="Green">E45</font></i></a>, <a class="ref" href="taylor_1.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E9:51_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E10:51_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:51"><i><font color="Green">E45</font></i></a>, <a class="ref" href="taylor_1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E11:51_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="fdiff_1.html#T33">FDIFF_1:33</a></i>;<br/><a NAME="E12:51_1_1_3"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E4:51_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/><a NAME="E13:51_1_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:51"><i><font color="Green">E45</font></i></a>, <a class="txt" href="taylor_1.html#E6:51_1_1_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="taylor_1.html#T31" target="_self">Th31</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E15:51_1_1_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>and </b><br/><a NAME="E16:51_1_1_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</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"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:51_1_1_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="taylor_1.html#E4:51_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="taylor_1.html#E8:51_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="taylor_1.html#E12:51_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="taylor_1.html#T27" target="_self">Th27</a></i>;<br/><i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:51"><i><font color="Green">E45</font></i></a>, <a class="txt" href="taylor_1.html#E6:51_1_1_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="taylor_1.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="taylor_1.html#E15:51_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E18:51_1_1_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:51_1_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><a NAME="E19:51_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="taylor_1.html#E15:51_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><a NAME="E20:51_1_1_3"/><b>then </b><i><font color="Green">E56</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 ;<br/><a NAME="E21:51_1_1_3"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E22:51_1_1_3"/><b>then </b>
0 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E18:51_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E23:51_1_1_3"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E24:51_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E20:51_1_1_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E25:51_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E18:51_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E26:51_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i><a class="txt" href="taylor_1.html#E18:51_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E18:51_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>

;<br/><b>hence </b><a NAME="E28:51_1_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</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"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E16:51_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="taylor_1.html#E17:51_1_1_3"><i><font color="Green">E54</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="E5:51"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K5" target="_self">Taylor</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</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"><span class="p5">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> )
 ;<br/>


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