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<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>for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">n</font>];<br/>
<a NAME="E1:23_1"/><i><font color="Green" title="E9">A2</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="nat_1.html#T2" target="_self" title="NAT_1:th.2">Th2</a></i>;<br/>
<a NAME="E2:23_1"/><i><font color="Green" title="E10">A3</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="23_1_1"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:23_1_1"/><i><font color="Green" title="E11">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:23_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/>

<a NAME="E3:23_1_1"/><b>then </b>
<font color="Maroon" title="c2">n</font> <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="nat_1.html#T8" target="_self" title="NAT_1:th.8">Th8</a></i>;<br/>
<a NAME="E4:23_1_1"/><b>then </b>
( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> or ( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c2">n</font> <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="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:23_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>]
 <b>by </b><i><a class="txt" href="nat_1.html#E1:23"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="nat_1.html#E1:23_1_1"><i><font color="Green" title="E11">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<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/>

<a NAME="E3:23_1"/>
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" target="_self" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="nat_1.html#E1:23_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="nat_1.html#E2:23_1"><i><font color="Green" title="E10">A3</font></i></a>);<br/></i>
<a NAME="E4:23_1"/><b>then </b>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
;<br/>
<b>hence </b><a NAME="E5:23_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 <b>by </b><i><a class="txt" href="nat_1.html#E1:23"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/>


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