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<b>let </b><font color="Maroon" title="c1">p</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"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> implies for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> )</span><br/>

<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E2">A1</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font>
 ; <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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span><br/>

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c2">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> )</span><br/>

<b>assume </b><a NAME="E2:4"/><i><font color="Green" title="E3">A2</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>suppose </b></a><a NAME="E1:4_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span><br/><div class="add"><b>hence </b><a NAME="E2:4_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
 ; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2"><b>suppose </b></a><a NAME="E1:4_1_2"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span><br/><div class="add"><b>thus </b><a NAME="E2:4_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:4_1_2_1"/><i><font color="Green" title="E5">A4</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="#E2:4"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="#E1:4_1_2"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_2_1_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:4_1_2_1_1_1_1"/><i><font color="Green" title="E6">A5</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> <font color="Maroon" title="c2">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span><br/><div class="add"><a NAME="E2:4_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:4"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></i>;<br/><b>hence </b><a NAME="E3:4_1_2_1_1_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="#E2:4"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="#E1:4_1_2"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="#E1:4_1_2_1_1_1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:4_1_2_1_1_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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"> <font color="Maroon" title="c2">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span><br/><div class="add"><b>hence </b><a NAME="E2:4_1_2_1_1_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="#E1:4"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="#E2:4"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="#E1:4_1_2_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="power.html#T42" title="POWER:th.42">POWER:42</a></i>; <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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:4_1_2_1"/>
<font color="Maroon" title="c2">a</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
 ; <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></div><b>end;</b></div></div><b>end;</b></div>

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