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<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/>

<a NAME="E1:18"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means 0 <a href="newton.html#K3" target="_self">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0;<br/>
<a NAME="E2:18"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a></i>;<br/>
<div><i><font color="Green">E11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:18_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] )
 ;<br/>0 <a href="newton.html#K3" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">0 <a href="newton.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 0
<b>by </b><i><a class="ref" href="newton.html#T11" target="_self">Th11</a></i>
<br/>.= 
0

;<br/><b>hence </b><a NAME="E3:18_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:18"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><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#S9">NAT_1:sch 9</a>(<a class="txt" href="newton.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="newton.html#E3:18"><i><font color="Green">E11</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:18"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 implies 0 <a href="newton.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="newton.html#E1:18"><i><font color="Green">E9</font></i></a></i>;<br/>


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