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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E22">A1</font></i>: 
not <font color="Maroon" title="c1">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/>

<a NAME="E2:33"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<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>not <font color="Maroon" title="c1">i</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> $1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>;<br/>
<a NAME="E3:33"/>
1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></i>;<br/>
<a NAME="E4:33"/><b>then </b><i><font color="Green" title="E24">A3</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="pepin.html#E2:33"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="nat_2.html#T24" title="NAT_2:th.24">NAT_2:24</a></i>;<br/>
<a NAME="E5:33"/><i><font color="Green" title="E25">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="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="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green" title="E26">A5</font></i>: 
not <font color="Maroon" title="c1">i</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/>

<a NAME="E2:33_1"/>
<font color="Maroon" title="c1">i</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/>
<b>hence </b><a NAME="E3:33_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="pepin.html#E1:33"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="pepin.html#E1:33_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="pepin.html#T19" target="_self" title="PEPIN:th.19">Th19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:33"/>
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" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="pepin.html#E4:33"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="pepin.html#E5:33"><i><font color="Green" title="E25">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:33"/>
not <font color="Maroon" title="c1">i</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/>


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