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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">k</font> <a href="radix_3.html#K3" title="RADIX_3:func.3">-SD_Sub_S</a> </span><br/><b>let </b><font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font> implies  <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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">k</font> <a href="radix_3.html#K3" title="RADIX_3:func.3">-SD_Sub_S</a>  )</span><br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">k</font> <a href="radix_3.html#K3" title="RADIX_3:func.3">-SD_Sub_S</a> </span><br/>

<a NAME="E2:25"/><b>then </b>
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:25"/><b>then </b>
<font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T16" title="XREAL_1:th.16">XREAL_1:16</a></i>;<br/>
<a NAME="E4:25"/><b>then </b><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></i>;<br/>
<a NAME="E5:25"/><b>then </b><i><font color="Green" title="E22">A3</font></i>: 
2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="power.html#T40" title="POWER:th.40">POWER:40</a></i>;<br/>
<a NAME="E6:25"/><i><font color="Green" title="E23">A4</font></i>: 
<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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="#E4:25"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="int_1.html#T79" title="INT_1:th.79">INT_1:79</a>, <a class="ref" href="power.html#T40" title="POWER:th.40">POWER:40</a></i>;<br/>
<a NAME="E7:25"/>
 <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="#E1:25"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="radix_3.html#T13" target="_self" title="RADIX_3:th.13">Th13</a></i>;<br/>
<a NAME="E8:25"/><b>then </b><i><font color="Green" title="E24">A5</font></i>: 
 <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> <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="#E6:25"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:25"/><i><font color="Green" title="E25">A6</font></i>: 
 <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>
 
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="radix_3.html#T13" target="_self" title="RADIX_3:th.13">Th13</a></i>;<br/>
<a NAME="E10:25"/>
 <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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 
<b>by </b><i><a class="txt" href="#E5:25"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E11:25"/><b>then </b><i><font color="Green" title="E26">A7</font></i>: 
 <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#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="c2">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E9:25"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:25"/>
 <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> is    <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>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E13:25"/>
 <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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">k</font> <a href="radix_3.html#K3" title="RADIX_3:func.3">-SD_Sub_S</a> 
 <b>by </b><i><a class="txt" href="#E8:25"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="#E11:25"><i><font color="Green" title="E26">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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