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<b>let </b><font color="Maroon" title="c1">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"> ( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">n</font> being   <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>  holds  <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,<font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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> being   <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>  holds  <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,<font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b><span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> $1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>;<br/>
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#T10" title="COMPLFLD:th.10">COMPLFLD:10</a>, <a class="ref" href="group_1.html#D8" title="GROUP_1:def.8">GROUP_1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
<b>by </b><i><a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a></i>
;<br/>
<a NAME="E3:81"/><b>then </b><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<div><i><font color="Green" title="E38">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><b>assume </b><a NAME="E1:81_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><b>then </b><span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
<b>by </b><i><a class="ref" href="group_1.html#D8" title="GROUP_1:def.8">GROUP_1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
<b>by </b><i><a class="txt" href="hahnban1.html#E1:81"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
;<br/><b>hence </b><a NAME="E3:81_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:81"/>
for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="hahnban1.html#E3:81"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="hahnban1.html#E4:81"><i><font color="Green" title="E38">A3</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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