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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a<sub>1</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>
<a NAME="E1:7_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E2:7_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:7_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>]
 ;<br/>

<b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:7_1_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1" target="_self">+</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>hence </b><a NAME="E4:7_1_1_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:7_1_1"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" target="_self">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:7_1_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div>
