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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</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">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 implies <font color="Maroon" title="c3">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">b</font> )</span><br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</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">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">b</font></span><br/>

<a NAME="E2:44"/><b>then </b><i><font color="Green" title="E30">A2</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</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> 
 
;<br/>
<a NAME="E3:44"/><b>then </b><i><font color="Green" title="E31">A3</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</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> 
 
<b>by </b><i><a class="ref" href="power.html#T39" target="_self" title="POWER:th.39">Th39</a></i>;<br/>
<a NAME="E4:44"/><i><font color="Green" title="E32">A4</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">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> 
 
<b>by </b><i><a class="txt" href="#E2:44"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="power.html#T39" target="_self" title="POWER:th.39">Th39</a></i>;<br/>
<a NAME="E5:44"/>
<font color="Maroon" title="c3">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</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> 
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="power.html#T39" target="_self" title="POWER:th.39">Th39</a></i>;<br/>
<a NAME="E6:44"/><b>then </b><i><font color="Green" title="E33">A5</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E7:44"/>
<font color="Maroon" title="c3">c</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font>
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></i>;<br/>
<a NAME="E8:44"/><b>then </b><i><font color="Green" title="E34">A6</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font>
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>;<br/>
<a NAME="E9:44"/>
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</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> 
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E10:44"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="#E8:44"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="power.html#T40" target="_self" title="POWER:th.40">Th40</a></i>;<br/>
<a NAME="E11:44"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="power.html#T34" target="_self" title="POWER:th.34">Th34</a></i>;<br/>
<a NAME="E12:44"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:44"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E13:44"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="#E4:44"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E14:44"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="power.html#T36" target="_self" title="POWER:th.36">Th36</a></i>;<br/>
<a NAME="E15:44"/><b>then </b>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="power.html#T31" target="_self" title="POWER:th.31">Th31</a></i>;<br/>
<a NAME="E16:44"/><b>then </b>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">1 <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="power.html#T36" target="_self" title="POWER:th.36">Th36</a></i>;<br/>
<a NAME="E17:44"/><b>then </b>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T31" target="_self" title="POWER:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E18:44"/>
<font color="Maroon" title="c3">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">c</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="#E6:44"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="xreal_1.html#T93" title="XREAL_1:th.93">XREAL_1:93</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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