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<b>let </b><font color="Maroon" title="c1">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>



<b>assume </b><a NAME="E1:9"/>
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="radix_3.html#K1" title="RADIX_3:func.1">-SD_Sub_S</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E3:9"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font>
 <b>and </b><br/><a NAME="E4:9"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0: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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:9"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/>
<a NAME="E6:9"/>
<font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/>
<a NAME="E7:9"/><b>then </b><i><font color="Green" title="E12">A4</font></i>: 
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></i>;<br/>
<a NAME="E8:9"/><b>then </b>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0: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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E9:9"/><b>then </b>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0: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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0: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"><span class="p3">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E10:9"/><b>then </b><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0: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"><span class="p3">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_3.html#E4:9"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:9"/>
<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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="radix_3.html#E7:9"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E12:9"/><b>then </b>
<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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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"><span class="p3">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E13:9"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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"><span class="p3">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="radix_3.html#E5:9"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E14:9"/>
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="radix_3.html#K1" title="RADIX_3:func.1">-SD_Sub_S</a> 
 <b>by </b><i><a class="txt" href="radix_3.html#E3:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="radix_3.html#E10:9"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>


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