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

<b>assume </b><a NAME="E1:86"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">x</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>;<br/>
<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">x</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>,0 = 
<a href="complex1.html#K6">1r</a>  <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K4">to_power</a> 0</span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
;<br/>
<a NAME="E3:86"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:86_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>then </b><span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">x</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">x</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K2">*</a> 0</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K4">to_power</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#T8" target="_self">Th8</a>, <a class="ref" href="power.html#T32">POWER:32</a></i>
;<br/><b>hence </b><a NAME="E3:86_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:86"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="hahnban1.html#T9" target="_self">Th9</a>, <a class="ref" href="hahnban1.html#T10" target="_self">Th10</a>);</i><br/>


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