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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="radix_1.html#K2" title="RADIX_1:func.2">-SD</a> </span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 <a href="radix_1.html#K2" title="RADIX_1:func.2">-SD</a> ;<br/>
<a NAME="E1:16"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="radix_1.html#T10" target="_self" title="RADIX_1:th.10">Th10</a></i>;<br/>
<a NAME="E2:16"/><i><font color="Green" title="E14">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">kk</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">kk</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green" title="E15">A3</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">kk</font> <a href="radix_1.html#K2" title="RADIX_1:func.2">-SD</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:16_1"/>
<font color="Maroon" title="c2">kk</font> <a href="radix_1.html#K2" title="RADIX_1:func.2">-SD</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="radix_1.html#K2" title="RADIX_1:func.2">-SD</a> 
 
<b>by </b><i><a class="ref" href="radix_1.html#T12" target="_self" title="RADIX_1:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="#E1:16_1"><i><font color="Green" title="E15">A3</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/>


</div><b>end;</b></div>
<a NAME="E3:16"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E1:16"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E2:16"><i><font color="Green" title="E14">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:16"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="radix_1.html#K2" title="RADIX_1:func.2">-SD</a> 
 ; <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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