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<span class="kw">let </span><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:()"><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> 1 &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">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font> implies  <a href="power.html#K5" title="POWER:func.5">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K5" title="POWER:func.5">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E2:61"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E3:61"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">and </span><br/><a NAME="E4:61"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
 <a href="power.html#K5" title="POWER:func.5">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K5" title="POWER:func.5">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:61"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="power.html#D3" target="_self" title="POWER:def.3">Def3</a></span>;<br/>
<a NAME="E6:61"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="power.html#D3" target="_self" title="POWER:def.3">Def3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:61_1_1"/>
(  <a href="power.html#K5" title="POWER:func.5">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="power.html#K5" title="POWER:func.5">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) or  <a href="power.html#K5" title="POWER:func.5">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="power.html#K5" title="POWER:func.5">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_1"/>
 <a href="power.html#K5" title="POWER:func.5">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="power.html#K5" title="POWER:func.5">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:61_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E6:61"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="power.html#T39" target="_self" title="POWER:th.39">Th44</a></span>; <a class="txt" 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="61_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_2"/>
 <a href="power.html#K5" title="POWER:func.5">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="power.html#K5" title="POWER:func.5">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:61_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E6:61"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="power.html#D3" target="_self" title="POWER:def.3">Def3</a></span>; <a class="txt" 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="E8:61"/>
contradiction
 ; <a class="txt" 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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