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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span><a href="sin_cos.html#K4" title="SIN_COS:func.4">Prod_real_n</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="newton.html#K11" title="NEWTON:func.11">!</a> ;<br/>
<a NAME="E1:9_1_1"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D5" target="_self" title="SIN_COS:def.5">Def5</a>, <a class="ref" href="newton.html#T18" title="NEWTON:th.18">NEWTON:18</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E5">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">l</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">l</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:9_1_1_1"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">l</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><span class="lab"><font color="Green" title="E7">A4</font></span>: <a href="sin_cos.html#K4" title="SIN_COS:func.4">Prod_real_n</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K4" title="SIN_COS:func.4">Prod_real_n</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D5" target="_self" title="SIN_COS:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">!</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="newton.html#T21" title="NEWTON:th.21">NEWTON:21</a></span>
;<br/><span class="kw">thus </span><a NAME="E3:9_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_1_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:9_1_1"/><span class="lab"><font color="Green" title="E6">A5</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E1:9_1_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E2:9_1_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>);<br/></span>
<span class="kw">thus </span><a NAME="E4:9_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a>  iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K4" title="SIN_COS:func.4">Prod_real_n</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_1"><span class="lab"><font color="Green" title="E6">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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