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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><font color="Maroon" title="c1">n</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>;<br/>
<a NAME="E1:9_1_1"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E2:9_1_1"/><i><font color="Green" title="E9">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="9_1_1_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>;<br/>

<a NAME="E1:9_1_1_1"/>
<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="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">kk</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">kk</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
;<br/>
<b>hence </b><a NAME="E2:9_1_1_1"/>
( <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] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:9_1_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" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="pepin.html#E1:9_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="pepin.html#E2:9_1_1"><i><font color="Green" title="E9">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:9_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>
 ;<br/>


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