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<b>let </b><font color="Maroon" title="c1">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E12">A1</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font>
 ;<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> ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">t</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">t</font> &amp; <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> );<br/>
<a NAME="E2:32"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E3:32"/><b>then </b><i><font color="Green" title="E13">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="txt" href="nat_1.html#E1:32"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>
<a NAME="E4:32"/><i><font color="Green" title="E14">A3</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<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>given </b><font color="Maroon" title="c3">k1</font>, <font color="Maroon" title="c4">t1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E2:32_1"/><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">t1</font> &amp; <font color="Maroon" title="c4">t1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> )
 ;<br/>

<a NAME="E3:32_1"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c4">t1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font>
 
<b>by </b><i><a class="txt" href="nat_1.html#E2:32_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="nat_1.html#T13" target="_self" title="NAT_1:th.13">Th13</a></i>;<br/>
<a NAME="E4:32_1"/><i><font color="Green" title="E17">A6</font></i>: 
( <font color="Maroon" title="c4">t1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> implies  ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">t</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">t</font> &amp; <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1_1"/><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c4">t1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font>
 ;<br/>

<b>take </b><font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c3">k1</font>;<br/>

<b>take </b><font color="Maroon" title="c6">t</font> = <font color="Maroon" title="c4">t1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>

<b>thus </b><a NAME="E2:32_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">t</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E2:32_1"><i><font color="Green" title="E15">A4</font></i></a></i>;<br/>

<b>thus </b><a NAME="E3:32_1_1"/>
<font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:32_1_1"><i><font color="Green" title="E18">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:32_1"/>
( <font color="Maroon" title="c4">t1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> implies  ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">t</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">t</font> &amp; <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1_2"/><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c4">t1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 ;<br/>

<b>take </b><font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>

<b>take </b><font color="Maroon" title="c6">t</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>

<b>thus </b><a NAME="E2:32_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">t</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E2:32_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="nat_1.html#E1:32_1_2"><i><font color="Green" title="E18">A8</font></i></a></i>;<br/>

<b>thus </b><a NAME="E3:32_1_2"/>
<font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:32"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:32_1"/>
 ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">t</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">t</font> &amp; <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> )
 <b>by </b><i><a class="txt" href="nat_1.html#E3:32_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="nat_1.html#E4:32_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:32"/>
for <font color="Olive" title="b1">n</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">n</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#E3:32"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="nat_1.html#E4:32"><i><font color="Green" title="E14">A3</font></i></a>);</i><br/>


</div>
