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<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:30"/><i><font color="Green">E31</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b> ex <font color="Olive">k</font>, <font color="Olive">t</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> );<br/>
<a NAME="E2:30"/>
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
 
;<br/>
<a NAME="E3:30"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E4:30"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>given </b><font color="Maroon">k1</font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">t1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:30_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">t1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:30_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">t1</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:30_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">t1</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> implies  ex <font color="Olive">k</font>, <font color="Olive">t</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">t1</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/>

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

<b>take </b><font color="Maroon">t</font> = <font color="Maroon">t1</font> <a href="xcmplx_0.html#K2">+</a> 1;<br/>

<b>thus </b><a NAME="E2:30_1_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E3:30_1_1"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:30_1"/>
( <font color="Maroon">t1</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> implies  ex <font color="Olive">k</font>, <font color="Olive">t</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">t1</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/>

<b>take </b><font color="Maroon">k</font> = <font color="Maroon">k1</font> <a href="xcmplx_0.html#K2">+</a> 1;<br/>

<b>take </b><font color="Maroon">t</font> = 0;<br/>

<b>thus </b><a NAME="E2:30_1_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E3:30_1_2"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:30_1"/>
 ex <font color="Olive">k</font>, <font color="Olive">t</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:30"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" target="_self">NAT_1:sch 2</a>(, );</i><br/>


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