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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 )
 </span><b>by </b><i><a class="ref" href="cqc_the1.html#T4">CQC_THE1:4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1"><b>suppose </b></a><a NAME="E1:80_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:80_1_1_1"/>
( <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 6 &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 4 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 24 )
 <b>by </b><i><a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_2"><b>suppose </b></a><a NAME="E1:80_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:80_1_1_2"/>
( <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 6 &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 4 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 24 )
 <b>by </b><i><a class="ref" href="newton.html#T31">NEWTON:31</a>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_3"><b>suppose </b></a><a NAME="E1:80_1_1_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:80_1_1_3"/>
( <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 6 &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 4 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 24 )
 <b>by </b><i><a class="ref" href="newton.html#T31">NEWTON:31</a>, <a class="ref" href="newton.html#T33">NEWTON:33</a>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_4"><b>suppose </b></a><a NAME="E1:80_1_1_4"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1, <font color="Maroon">n2</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/><a NAME="E3:80_1_1_4"/>
<font color="Maroon">n2</font> <a href="newton.html#K12">!</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T23">NEWTON:23</a></i>;<br/><a NAME="E4:80_1_1_4"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><a NAME="E5:80_1_1_4"/>
( <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>;<br/><a NAME="E6:80_1_1_4"/><b>then </b>
<font color="Maroon">n</font> <a href="newton.html#K12">!</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>
 ;<br/><b>then </b><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="stirl2_1.html#D1" target="_self">Def1</a>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
<b>by </b><i><a class="ref" href="xcmplx_1.html#T84">XCMPLX_1:84</a>, <a class="ref" href="newton.html#T20">NEWTON:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
<b>by </b><i><a class="ref" href="stirl2_1.html#T15" target="_self">Th15</a></i>
;<br/><b>hence </b><a NAME="E8:80_1_1_4"/>
( <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 6 &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 4 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 24 )
 <b>by </b><i><a class="ref" href="stirl2_1.html#D1" target="_self">Def1</a>, <a class="ref" href="newton.html#T31">NEWTON:31</a>, <a class="ref" href="newton.html#T33">NEWTON:33</a>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_5"><b>suppose </b></a><a NAME="E1:80_1_1_5"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3
 ;<br/><div class="add"><a NAME="E2:80_1_1_5"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1, <font color="Maroon">n2</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2, <font color="Maroon">n3</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 3, <font color="Maroon">n4</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 4 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T61">NAT_1:61</a></i>;<br/><a NAME="E4:80_1_1_5"/>
( <font color="Maroon">n2</font> <a href="newton.html#K12">!</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n3</font> <a href="newton.html#K12">!</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n4</font> <a href="newton.html#K12">!</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="newton.html#T23">NEWTON:23</a></i>;<br/><a NAME="E5:80_1_1_5"/><b>then </b><i><font color="Green">E47</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">n4</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">n4</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><a NAME="E6:80_1_1_5"/><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">n4</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n4</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">n4</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>;<br/><a NAME="E7:80_1_1_5"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="stirl2_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then </b><i><font color="Green">E62</font></i>: <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="stirl2_1.html#T16" target="_self">Th16</a>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 2
<b>by </b><i><a class="ref" href="xcmplx_1.html#T84">XCMPLX_1:84</a>, <a class="ref" href="newton.html#T20">NEWTON:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n1</font></span>)</span> <a href="binop_2.html#K12">/</a> 2
<b>by </b><i><a class="ref" href="stirl2_1.html#T15" target="_self">Th15</a></i>
;<br/><i><font color="Green">E63</font></i>: <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 3 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n3</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n1</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n3</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">3 <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="stirl2_1.html#D1" target="_self">Def1</a>, <a class="ref" href="stirl2_1.html#T16" target="_self">Th16</a>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n3</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">n3</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n1</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 6
<b>by </b><i><a class="ref" href="xcmplx_1.html#T84">XCMPLX_1:84</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n1</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">n2</font></span>)</span> <a href="binop_2.html#K12">/</a> 6
<b>by </b><i><a class="ref" href="stirl2_1.html#T15" target="_self">Th15</a></i>
;<br/><a NAME="E10:80_1_1_5"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="stirl2_1.html#D1" target="_self">Def1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><i><font color="Green">E65</font></i>: <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 4 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n4</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n3</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n2</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">n1</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n4</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">4 <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="stirl2_1.html#T16" target="_self">Th16</a>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n4</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">n4</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n3</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n2</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">n1</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 24
<b>by </b><i><a class="ref" href="xcmplx_1.html#T84">XCMPLX_1:84</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n1</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">n2</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">n3</font></span>)</span> <a href="binop_2.html#K12">/</a> 24
<b>by </b><i><a class="ref" href="stirl2_1.html#T15" target="_self">Th15</a></i>
;<br/><a NAME="E12:80_1_1_5"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="stirl2_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E13:80_1_1_5"/>
( <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 6 &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 4 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 24 )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T17" target="_self">Th17</a>, <a class="ref" href="stirl2_1.html#T18" target="_self">Th18</a>, , <a class="ref" href="newton.html#T33">NEWTON:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:80"/>
( <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 6 &amp; <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> 4 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 24 )
 ;<br/>


</div>
