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<div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1"><b>suppose </b></a><a NAME="E1:2_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T54" title="PREPOWER:th.54">PREPOWER:54</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2"><b>suppose </b></a><a NAME="E1:2_1_2"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>thus </b><a NAME="E2:2_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1"><b>proof </b></a><div class="add">

<i><font color="Green" title="E4">A2</font></i>: 0 <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> = 
0 <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D4" title="PREPOWER:def.4">PREPOWER:def 4</a></i>
<br/>.= 
0 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">0 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">m</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_2_1_2"/>
( ( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) or <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or <font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2_1_2_1"><b>suppose </b></a><a NAME="E1:2_1_2_1_2_1"/><i><font color="Green" title="E5">A3</font></i>: 
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><div class="add"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0</span>)</span> = 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> 0</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> 0</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T44" title="PREPOWER:th.44">PREPOWER:44</a></i>
;<br/><b>hence </b><a NAME="E3:2_1_2_1_2_1"/>
<font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:2_1_2_1_2_1"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2_1_2_2"><b>suppose </b></a><a NAME="E1:2_1_2_1_2_2"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:2_1_2_1_2_2"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">n</font>
 ;<br/><a NAME="E3:2_1_2_1_2_2"/><b>then </b><i><font color="Green" title="E5">A4</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:2_1_2_1_2_2"/><b>then </b><i><font color="Green" title="E6">A5</font></i>: 
0 <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">0 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:2_1_2_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>;<br/><span class="p1">(<span class="default">0 <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">0 <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <font color="Maroon" title="c2">m</font></span>)</span> = 
<span class="p1">(<span class="default">0 <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">0 <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <font color="Maroon" title="c2">m</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D4" title="PREPOWER:def.4">PREPOWER:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">0 <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <font color="Maroon" title="c2">m</font></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">0 <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <font color="Maroon" title="c2">m</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:2_1_2_1_2_2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>
;<br/><b>hence </b><a NAME="E6:2_1_2_1_2_2"/>
<font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:2_1_2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="taylor_1.html#E4:2_1_2_1_2_2"><i><font color="Green" title="E6">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2_1_2_3"><b>suppose </b></a><a NAME="E1:2_1_2_1_2_3"/>
<font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:2_1_2_1_2_3"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">m</font>
 ;<br/><a NAME="E3:2_1_2_1_2_3"/><b>then </b><i><font color="Green" title="E5">A6</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:2_1_2_1_2_3"/><b>then </b><i><font color="Green" title="E6">A7</font></i>: 
0 <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">0 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:2_1_2_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>;<br/><span class="p1">(<span class="default">0 <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">0 <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <font color="Maroon" title="c2">m</font></span>)</span> = 
<span class="p1">(<span class="default">0 <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">0 <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D4" title="PREPOWER:def.4">PREPOWER:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">0 <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">0 <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:2_1_2_1_2_3"><i><font color="Green" title="E5">A6</font></i></a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>
;<br/><b>hence </b><a NAME="E6:2_1_2_1_2_3"/>
<font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:2_1_2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="taylor_1.html#E4:2_1_2_1_2_3"><i><font color="Green" title="E6">A7</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
