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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">c</font>, <font color="Maroon" title="c3">d</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> implies <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:50"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E3:50"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:50_1_1"/>
( <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">d</font> or <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">d</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:50_1_1_1"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">d</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font></span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:50_1_1_1_1_1"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:50_1_1_1_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:50_1_1_1_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E1:50_1_1_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="power.html#T40" target="_self" onmouseover="rs('power/T40')" onmouseout="rh()">Th40</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:50_1_1_1_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:50_1_1_1_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="power.html#T39" target="_self" onmouseover="rs('power/T39')" onmouseout="rh()">Th39</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:50_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:50_1_1_2"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">d</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font></span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:50_1_1_2_1_1"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:50_1_1_2_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:50_1_1_2_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E1:50_1_1_2"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="power.html#T40" target="_self" onmouseover="rs('power/T40')" onmouseout="rh()">Th40</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:50_1_1_2_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:50_1_1_2_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1_2"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="power.html#T39" target="_self" onmouseover="rs('power/T39')" onmouseout="rh()">Th39</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:50_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:50"/>
<font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c3">d</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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