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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E1:95"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1</span><br/>

<a NAME="E2:95"/><span class="kw">then </span>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></span>;<br/>
<a NAME="E3:95"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></span>;<br/>
<a NAME="E4:95"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:95"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:95"/>
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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