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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E13">nk</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">10 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b> ex <font color="Olive" title="b1">k</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">10 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> $1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp; 9 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">k</font> );<br/>
<i><font color="Green" title="E14">dz</font></i>: <span class="p1">(<span class="default">10 <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> 0</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 = 
1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>
<br/>.= 
0

;<br/>
<a NAME="E3:12"/>
9 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 0
 
<b>by </b><i><a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a></i>;<br/>
<a NAME="E4:12"/><b>then </b><i><font color="Green" title="E15">z</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="numeral1.html#E2:12"><i><font color="Green" title="E14">dz</font></i></a></i>;<br/>
<div><i><font color="Green" title="E16">k</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">k</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ;<br/><b>then consider </b><font color="Maroon" title="c4">l</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green" title="E17">l</font></i>: 
( <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">10 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp; 9 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">l</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c5">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E5:12_2"/><i><font color="Green" title="E18">m</font></i>: 
<font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a>, <a class="txt" href="numeral1.html#E3:12_2"><i><font color="Green" title="E17">l</font></i></a></i>;<br/><i><font color="Green" title="E19">dz</font></i>: <span class="p1">(<span class="default">10 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 10</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a>, <a class="txt" href="numeral1.html#E5:12_2"><i><font color="Green" title="E18">m</font></i></a>, <a class="txt" href="numeral1.html#E3:12_2"><i><font color="Green" title="E17">l</font></i></a></i>
<br/>.= 
9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 10</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>

;<br/><a NAME="E7:12_2"/>
9 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 10</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>hence </b><a NAME="E8:12_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="numeral1.html#E6:12_2"><i><font color="Green" title="E19">dz</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:12"/>
for <font color="Olive" title="b1">k</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="numeral1.html#E4:12"><i><font color="Green" title="E15">z</font></i></a>, <a class="txt" href="numeral1.html#E5:12"><i><font color="Green" title="E16">k</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c3">l</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E8:12"/><i><font color="Green" title="E17">l</font></i>: 
( <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">10 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp; 9 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">l</font> )
 ;<br/>
<b>thus </b><a NAME="E9:12"/>
9 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:12"><i><font color="Green" title="E13">nk</font></i></a>, <a class="txt" href="numeral1.html#E8:12"><i><font color="Green" title="E17">l</font></i></a></i>;<br/>


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