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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="newton.html#K6" target="_self">!</a>  is   <a href="nat_1.html#NM1">Nat</a>;<br/>
<a NAME="E1:25"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_2.html#T58">FINSEQ_2:58</a>, <a class="ref" href="rvsum_1.html#T124">RVSUM_1:124</a></i>;<br/>
<div><i><font color="Green">E14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K6" target="_self">!</a>  as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:25_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K6" target="_self">!</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="newton.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E4:25_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:25"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(<a class="txt" href="newton.html#E1:25"><i><font color="Green">E13</font></i></a>, <a class="txt" href="newton.html#E2:25"><i><font color="Green">E14</font></i></a>);</i><br/>


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