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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>( <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> $1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">$1 <a href="newton.html#K6" title="NEWTON:func.6">!</a> </span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c5">s1</font>, <font color="Maroon" title="c6">s2</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:35_1_2"/><i><font color="Green" title="E27">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c5">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>and </b><br/><a NAME="E2:35_1_2"/><i><font color="Green" title="E28">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c6">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_2_1"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">n</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="taylor_1.html#E1:35_1_2_1"><i><font color="Green" title="E29">A4</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c5">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">n</font>)
<b>by </b><i><a class="txt" href="taylor_1.html#E1:35_1_2"><i><font color="Green" title="E27">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:35_1_2"><i><font color="Green" title="E28">A3</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:35_1_2"/>
<font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T8" title="SEQ_1:th.8">SEQ_1:8</a></i>;<br/>


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