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<b>let </b><font color="Maroon" title="c1">x0</font>, <font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">Z</font> being <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">Z</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">m</font> holds <br/>( <span class="p1">(<span class="default"><a href="taylor_1.html#K6" title="TAYLOR_1:func.6">Taylor</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Olive" title="b3">Z</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Olive" title="b2">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K6" title="TAYLOR_1:func.6">Taylor</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Olive" title="b3">Z</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Olive" title="b1">n</font> )</span><br/><b>let </b><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Z</font> being <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">m</font> holds <br/>( <span class="p1">(<span class="default"><a href="taylor_1.html#K6" title="TAYLOR_1:func.6">Taylor</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Olive" title="b1">Z</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K6" title="TAYLOR_1:func.6">Taylor</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Olive" title="b1">Z</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">n</font> )</span><br/><b>let </b><font color="Maroon" title="c5">Z</font> be  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">m</font> implies ( <span class="p1">(<span class="default"><a href="taylor_1.html#K6" title="TAYLOR_1:func.6">Taylor</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K6" title="TAYLOR_1:func.6">Taylor</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">n</font> ) )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:43"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font>
 <b>and </b><br/><a NAME="E2:43"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="taylor_1.html#K6" title="TAYLOR_1:func.6">Taylor</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K6" title="TAYLOR_1:func.6">Taylor</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">n</font> )</span><br/>

<a NAME="E3:43"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a></i>;<br/>
<a NAME="E4:43"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a></i>;<br/>
<i><font color="Green" title="E30">A5</font></i>: <span class="p1">(<span class="default"><a href="taylor_1.html#K6" title="TAYLOR_1:func.6">Taylor</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <span class="p5">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c5">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D7" title="TAYLOR_1:def.7">TAYLOR_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="hfdiff_1.html#T36" target="_self" title="HFDIFF_1:th.36">Th27</a>, <a class="txt" href="#E2:43"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="#E1:43"><i><font color="Green" title="E26">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">m</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:43"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K6" title="TAYLOR_1:func.6">Taylor</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <span class="p5">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c5">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D7" title="TAYLOR_1:def.7">TAYLOR_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="hfdiff_1.html#T37" target="_self" title="HFDIFF_1:th.37">Th28</a>, <a class="txt" href="#E1:43"><i><font color="Green" title="E26">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">n</font>
<b>by </b><i><a class="txt" href="#E4:43"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>
;<br/>
<b>hence </b><a NAME="E7:43"/>
( <span class="p1">(<span class="default"><a href="taylor_1.html#K6" title="TAYLOR_1:func.6">Taylor</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K6" title="TAYLOR_1:func.6">Taylor</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">n</font> )
 <b>by </b><i><a class="txt" href="#E5:43"><i><font color="Green" title="E30">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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