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<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>let </b><font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:23"/>
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1"/>
( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> or (  <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="c2">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>suppose </b></a><a NAME="E1:23_1_1"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:23_1_1"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">k</font> &amp;  <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="radix_3.html#E1:23_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="radix_3.html#D3" target="_self" title="RADIX_3:def.3">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2"><b>suppose </b></a><a NAME="E1:23_1_2"/>
(  <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="c2">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:23_1_2"/><b>then </b>
 <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="radix_3.html#D3" target="_self" title="RADIX_3:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E3:23_1_2"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">k</font> &amp;  <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_3"><b>suppose </b></a><a NAME="E1:23_1_3"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:23_1_3"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">k</font> &amp;  <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="radix_3.html#E1:23_1_3"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="radix_3.html#D3" target="_self" title="RADIX_3:def.3">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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