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<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><a NAME="E1:54"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_1"/>
( <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> or <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">d</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 1 or <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">d</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_1_1_1_2"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_1_1_1_1_2"/>
<font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">d</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:54_1_1_1"/>
<font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">d</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">d</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 1 or <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">d</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_2_1_1_2"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_1_2_1_1_2"/>
<font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">d</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:54_1_1_2"/>
<font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">d</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:54"/>
<font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">d</font>
 ;<br/>


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