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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</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="c3">n</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> ;<br/>





<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E10">A1</font></i>: 
( ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</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>  &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) or ( <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  ex <font color="Olive" title="b1">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>  st <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1"/>
( ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</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>  &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) or ( <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  ex <font color="Olive" title="b1">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>  st <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="power.html#E1:15"><i><font color="Green" title="E10">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1"/><i><font color="Green" title="E11">A2</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</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>  &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:15_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="real_2.html#T125" title="REAL_2:th.125">REAL_2:125</a></i>;<br/><b>hence </b><font color="Maroon" title="c3">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span> = 
<font color="Maroon" title="c3">n</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span>
<b>by </b><i><a class="txt" href="power.html#E1:15_1_1_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="power.html#D1" target="_self" title="POWER:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c2">b</font></span>)</span>
<b>by </b><i><a class="txt" href="power.html#E1:15_1_1_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="prepower.html#T33" title="PREPOWER:th.33">PREPOWER:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c2">b</font></span>)</span>
<b>by </b><i><a class="txt" href="power.html#E1:15_1_1_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="power.html#D1" target="_self" title="POWER:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c2">b</font></span>)</span>
<b>by </b><i><a class="txt" href="power.html#E1:15_1_1_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="power.html#D1" target="_self" title="POWER:def.1">Def1</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_2"/><i><font color="Green" title="E11">A3</font></i>: 
( <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  ex <font color="Olive" title="b1">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>  st <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><div class="add"><b>thus </b><font color="Maroon" title="c3">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span> = 
<font color="Maroon" title="c3">n</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100" title="XCMPLX_1:th.100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="power.html#E1:15_1_1_2"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="power.html#T12" target="_self" title="POWER:th.12">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="power.html#E1:15_1_1_2"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="power.html#T13" target="_self" title="POWER:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c2">b</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100" title="XCMPLX_1:th.100">XCMPLX_1:100</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:15"/>
<font color="Maroon" title="c3">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c2">b</font></span>)</span>
 ;<br/>


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