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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#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> 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#K5" title="NUMBERS:func.5">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#K3" title="POWER:func.3">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#K3" title="POWER:func.3">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="E1:58"/><span class="lab"><font color="Green" title="E35">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="E2:58"/><span class="lab"><font color="Green" title="E36">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#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E3:58"/><span class="lab"><font color="Green" title="E37">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="E4:58"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
 <a href="power.html#K3" title="POWER:func.3">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#K3" title="POWER:func.3">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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"> contradiction</span><br/>

<a NAME="E5:58"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K3" title="POWER:func.3">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="#E1:58"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="power.html#D3" target="_self" onmouseover="rs('power/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E6:58"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K3" title="POWER:func.3">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="#E1:58"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="power.html#D3" target="_self" onmouseover="rs('power/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_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"> contradiction</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_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="E1:58_1_1"/>
(  <a href="power.html#K3" title="POWER:func.3">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#K3" title="POWER:func.3">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) or  <a href="power.html#K3" title="POWER:func.3">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#K3" title="POWER:func.3">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="#E4:58"><span class="lab"><font color="Green" title="E38">A4</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="58_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1"/>
 <a href="power.html#K3" title="POWER:func.3">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#K3" title="POWER:func.3">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E5:58"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E40">A6</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_2"/>
 <a href="power.html#K3" title="POWER:func.3">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#K3" title="POWER:func.3">log</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="power.html#D3" target="_self" onmouseover="rs('power/D3')" onmouseout="rh()">Def3</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="E8:58"/>
contradiction
 ; <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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