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<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> implies  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><br/>

<span class="kw">then </span> <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font> = 
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></span>
<br/>.= 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<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="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E3:1"/>
 <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</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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