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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 6 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 24 )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:65_1_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:65_1_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 6 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 24 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:65_1_1_1"/>
( <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 6 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 24 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#D3" title="NEWTON:def.3">NEWTON:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:65_1_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 6 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 24 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:65_1_1_2"/>
( <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 6 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 24 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T21" title="NEWTON:th.21">NEWTON:21</a>, <a class="ref" href="newton.html#D3" title="NEWTON:def.3">NEWTON:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:65_1_1_3"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 6 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 24 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:65_1_1_3"/>
( <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 6 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 24 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T21" title="NEWTON:th.21">NEWTON:21</a>, <a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a>, <a class="ref" href="newton.html#D3" title="NEWTON:def.3">NEWTON:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:65_1_1_4"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 6 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 24 )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">n1</font> = <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1, <font color="Maroon" title="c3">n2</font> = <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/><a NAME="E3:65_1_1_4"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">!</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></span>;<br/><a NAME="E4:65_1_1_4"/>
<font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T17" title="NEWTON:th.17">NEWTON:17</a></span>;<br/><a NAME="E5:65_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/><a NAME="E6:65_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">!</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_4"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E3:65_1_1_4"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="newton.html#D3" title="NEWTON:def.3">NEWTON:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K17" title="BINOP_2:func.17">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T14" title="NEWTON:th.14">NEWTON:14</a>, <a class="ref" href="xcmplx_1.html#T83" title="XCMPLX_1:th.83">XCMPLX_1:83</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:65_1_1_4"><span class="lab"><font color="Green" title="E46">A3</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E8:65_1_1_4"/>
( <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 6 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 24 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_4"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="newton.html#T21" title="NEWTON:th.21">NEWTON:21</a>, <a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a>, <a class="ref" href="newton.html#D3" title="NEWTON:def.3">NEWTON:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:65_1_1_5"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 6 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 24 )</span><br/><div class="add"><a NAME="E2:65_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">n1</font> = <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1, <font color="Maroon" title="c3">n2</font> = <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2, <font color="Maroon" title="c4">n3</font> = <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 3, <font color="Maroon" title="c5">n4</font> = <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 4 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:65_1_1_5"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">!</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></span>;<br/><a NAME="E5:65_1_1_5"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">!</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></span>;<br/><a NAME="E6:65_1_1_5"/>
<font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T17" title="NEWTON:th.17">NEWTON:17</a></span>;<br/><a NAME="E7:65_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/><a NAME="E8:65_1_1_5"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_5"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A8</font></span>: <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65_1_1_5"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E5:65_1_1_5"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="newton.html#D3" title="NEWTON:def.3">NEWTON:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K17" title="BINOP_2:func.17">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T14" title="NEWTON:th.14">NEWTON:14</a>, <a class="ref" href="xcmplx_1.html#T83" title="XCMPLX_1:th.83">XCMPLX_1:83</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n1</font></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:65_1_1_5"><span class="lab"><font color="Green" title="E47">A7</font></span></a></span>
;<br/><a NAME="E10:65_1_1_5"/>
<font color="Maroon" title="c5">n4</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T17" title="NEWTON:th.17">NEWTON:17</a></span>;<br/><a NAME="E11:65_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n4</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n4</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/><a NAME="E12:65_1_1_5"/><span class="lab"><font color="Green" title="E50">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n4</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">!</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n4</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n4</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></span>;<br/><a NAME="E13:65_1_1_5"/>
<font color="Maroon" title="c4">n3</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T17" title="NEWTON:th.17">NEWTON:17</a></span>;<br/><a NAME="E14:65_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">n3</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n3</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/><a NAME="E15:65_1_1_5"/><span class="lab"><font color="Green" title="E52">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">n3</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">!</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n3</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n3</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A13</font></span>: <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 3 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n3</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n3</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default">3 <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_5"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E4:65_1_1_5"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E5:65_1_1_5"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="newton.html#D3" title="NEWTON:def.3">NEWTON:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n3</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n3</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K17" title="BINOP_2:func.17">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 6
<span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#T50" target="_self" title="STIRL2_1:th.50">Th60</a>, <a class="ref" href="xcmplx_1.html#T83" title="XCMPLX_1:th.83">XCMPLX_1:83</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 6
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:65_1_1_5"><span class="lab"><font color="Green" title="E51">A11</font></span></a></span>
;<br/><a NAME="E17:65_1_1_5"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_5"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 4 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n4</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n3</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n4</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default">4 <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65_1_1_5"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E5:65_1_1_5"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E15:65_1_1_5"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E12:65_1_1_5"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="newton.html#D3" title="NEWTON:def.3">NEWTON:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n4</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n4</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></span>)</span></span>)</span> <a href="binop_2.html#K17" title="BINOP_2:func.17">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n3</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 24
<span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#T50" target="_self" title="STIRL2_1:th.50">Th60</a>, <a class="ref" href="xcmplx_1.html#T83" title="XCMPLX_1:th.83">XCMPLX_1:83</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">n3</font></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 24
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:65_1_1_5"><span class="lab"><font color="Green" title="E49">A9</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E19:65_1_1_5"/>
( <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 6 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 24 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_5"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E9:65_1_1_5"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="txt" href="#E16:65_1_1_5"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:65"/>
( <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 6 &amp; <font color="Maroon" title="c1">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 24 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
