<?xml version="1.0"?>
<div class="add">

<a NAME="E1:4"/><i><font color="Green">E38</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="binop_2.html#K12">/</a> 2 &amp; <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 0 )
 
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">k</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="newton.html#K9">choose</a> <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="newton.html#K9">choose</a> <font color="Olive">k</font>;<br/>
<a NAME="E2:4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<b>assume </b><a NAME="E2:4_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default">0 <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 ;<br/>

<a NAME="E3:4_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="int_1.html#T47">INT_1:47</a></i>;<br/>
<a NAME="E4:4_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:4_1"/>
0 <a href="newton.html#K9">choose</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="newton.html#K9">choose</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#T47">INT_1:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:4"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<a NAME="E2:4_2"/>
for <font color="Olive">k</font>, <font color="Olive">m1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m1</font> &amp; <font color="Olive">m1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><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#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Olive">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:4_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 ;<br/>

<b>assume </b><a NAME="E2:4_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><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#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 ;<br/>

<a NAME="E3:4_2_1"/><b>then </b>
( <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m1</font> )
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E4:4_2_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">m1</font> <a href="binop_2.html#K24">*</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E5:4_2_1"/>
1 <a href="binop_2.html#K24">*</a> <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m1</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E6:4_2_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>set </b><font color="Maroon">l1</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">m1</font>;<br/>
<a NAME="E7:4_2_1"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">m1</font>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>set </b><font color="Maroon">m</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>;<br/>
<b>set </b><font color="Maroon">l</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T80">INT_1:80</a></i>;<br/>
<a NAME="E9:4_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E10:4_2_1"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><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#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E11:4_2_1"/>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E12:4_2_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/>
<a NAME="E13:4_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E14:4_2_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E16:4_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:4_2_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_2_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:4_2_1_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:4_2_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:4_2_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:4_2_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:4_2_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:4_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/><a NAME="E7:4_2_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><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#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E8:4_2_1_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>set </b><font color="Maroon">nk</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font>;<br/>
<a NAME="E17:4_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>reconsider </b><font color="Maroon">nk</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">nk1</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font>;<br/>
<a NAME="E19:4_2_1"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1
 
;<br/>
<b>reconsider </b><font color="Maroon">nk1</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E21:4_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">n</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/>
<i><font color="Green">E70</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> 1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/>
<a NAME="E23:4_2_1"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T57">XCMPLX_1:57</a></i>;<br/>
<a NAME="E24:4_2_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">k</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="E25:4_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">nk</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/>
<i><font color="Green">E74</font></i>: <span class="p1">(<span class="default"><font color="Maroon">nk1</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">nk</font> <a href="newton.html#K12">!</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">nk</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">nk</font> <a href="newton.html#K12">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nk</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">nk</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> 1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/>
<a NAME="E27:4_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nk</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">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>;<br/>
<a NAME="E28:4_2_1"/>
0 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E29:4_2_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E30:4_2_1"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font> <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="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nk1</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">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>;<br/>
<i><font color="Green">E78</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nk</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</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="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nk1</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <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">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">nk</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nk1</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">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T83">XCMPLX_1:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">nk</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nk1</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">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T79">XCMPLX_1:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nk</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nk1</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">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T79">XCMPLX_1:79</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nk</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nk1</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T83">XCMPLX_1:83</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nk</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">nk1</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</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">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T85">XCMPLX_1:85</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nk</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">nk1</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T57">XCMPLX_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T85">XCMPLX_1:85</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>

;<br/>
<a NAME="E32:4_2_1"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">nk1</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">k</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T25">NEWTON:25</a></i>;<br/>
<a NAME="E33:4_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <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="E34:4_2_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T50">XCMPLX_1:50</a></i>;<br/>
<a NAME="E35:4_2_1"/><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</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="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T83">XCMPLX_1:83</a>, </i>;<br/>
<i><font color="Green">E82</font></i>: <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> 1

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/>
<a NAME="E37:4_2_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E38:4_2_1"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E39:4_2_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E40:4_2_1"/><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K9">+</a> 1
 
;<br/>
<a NAME="E41:4_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E42:4_2_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<i><font color="Green">E86</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T76">XCMPLX_1:76</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</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#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
;<br/>
<i><font color="Green">E87</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T76">XCMPLX_1:76</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</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#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_8"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_1_8"/>
( <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_8_1"><b>suppose </b></a><a NAME="E1:4_2_1_8_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">n</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:4_2_1_8_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font>
 <b>by </b><i>, <a class="ref" href="abian.html#D2">ABIAN:def 2</a></i>;<br/><i><font color="Green">E91</font></i>: <span class="p1"><a href="int_1.html#K1">[\</a><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#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="nat_4.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">k1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#T51">INT_1:51</a></i>
<br/>.= 
<font color="Maroon">k1</font> <a href="binop_2.html#K23">+</a> 0
<b>by </b><i>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="nat_4.html#T2" target="_self">Th2</a>, <a class="ref" href="int_1.html#T47">INT_1:47</a></i>
;<br/><i><font color="Green">E92</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">m1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">m1</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">l</font> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="nat_4.html#T3" target="_self">Th3</a>, </i>
;<br/><a NAME="E6:4_2_1_8_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_4.html#T3" target="_self">Th3</a>, , </i>;<br/><i><font color="Green">E94</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">m1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m1</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">m1</font></span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i>, , <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"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </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">m</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">l</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a>, <a class="ref" href="nat_4.html#T3" target="_self">Th3</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#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </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">m</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">l</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</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#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </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">l</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T85">XCMPLX_1:85</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T81">XCMPLX_1:81</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>
;<br/><a NAME="E8:4_2_1_8_1"/><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:4"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_4.html#T3" target="_self">Th3</a>, , , , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E9:4_2_1_8_1"/>
 <a href="binop_2.html#K7">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i>, <a class="ref" href="nat_4.html#T3" target="_self">Th3</a>, , <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E10:4_2_1_8_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:4_2_1_8_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_4.html#T1" target="_self">Th1</a>, <a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/><a NAME="E12:4_2_1_8_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="nat_4.html#T1" target="_self">Th1</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><a NAME="E13:4_2_1_8_1"/><b>then </b>
<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#K10">-</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</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> <span class="p4"><a href="int_1.html#K1">[\</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E14:4_2_1_8_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, , <a class="ref" href="nat_4.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_8_2"><b>suppose </b></a><a NAME="E1:4_2_1_8_2"/><i><font color="Green">E96</font></i>: 
not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">j</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:4_2_1_8_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="abian.html#T1">ABIAN:1</a></i>;<br/><i><font color="Green">E99</font></i>: <span class="p1"><a href="int_1.html#K1">[\</a><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#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="nat_4.html#T2" target="_self">Th2</a>, <a class="ref" href="int_1.html#T47">INT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="int_1.html#T51">INT_1:51</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="nat_4.html#T2" target="_self">Th2</a></i>
;<br/><i><font color="Green">E100</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">m1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">m1</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">l</font>
<b>by </b><i><a class="ref" href="nat_4.html#T3" target="_self">Th3</a>, </i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_8_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_1_8_2_3"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 or <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i>, , <a class="ref" href="nat_4.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_8_2_3_1"><b>suppose </b></a><a NAME="E1:4_2_1_8_2_3_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_2_1_8_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_4.html#T3" target="_self">Th3</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_8_2_3_2"><b>suppose </b></a><a NAME="E1:4_2_1_8_2_3_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:4_2_1_8_2_3_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:4_2_1_8_2_3_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><i><font color="Green">E103</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">m1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m1</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">m1</font></span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i>, , <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"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a>, <a class="ref" href="nat_4.html#T3" target="_self">Th3</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#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</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#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </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">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T85">XCMPLX_1:85</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T81">XCMPLX_1:81</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>
;<br/><a NAME="E5:4_2_1_8_2_3_2"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nk</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E2:4"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, </i>;<br/><a NAME="E6:4_2_1_8_2_3_2"/>
<font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E7:4_2_1_8_2_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K3">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E8:4_2_1_8_2_3_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E9:4_2_1_8_2_3_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E10:4_2_1_8_2_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E11:4_2_1_8_2_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E12:4_2_1_8_2_3_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/><a NAME="E13:4_2_1_8_2_3_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><a NAME="E14:4_2_1_8_2_3_2"/><b>then </b>
<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#K10">-</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E15:4_2_1_8_2_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:4"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:4_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:4"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E5:4"/>
for <font color="Olive">n</font>, <font color="Olive">k</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> holds <br/><font color="Olive">n</font> <a href="newton.html#K9">choose</a> <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> <a href="newton.html#K9">choose</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:4_3"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<b>assume </b><a NAME="E2:4_3"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_3_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_1_1"><b>suppose </b></a><a NAME="E1:4_3_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_3_1_1"/>
<font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_1_2"><b>suppose </b></a><a NAME="E1:4_3_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">r</font> = <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font>;<br/><a NAME="E2:4_3_1_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a>, </i>;<br/><a NAME="E3:4_3_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="newton.html#T30">NEWTON:30</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_3_1_2_1"/>
( <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_1_2_1_1"><b>suppose </b></a><a NAME="E1:4_3_1_2_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">n</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:4_3_1_2_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="abian.html#D2">ABIAN:def 2</a></i>;<br/><a NAME="E4:4_3_1_2_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i>, <a class="ref" href="int_1.html#T47">INT_1:47</a></i>;<br/><a NAME="E5:4_3_1_2_1_1"/>
 <a href="binop_2.html#K7">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E6:4_3_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E7:4_3_1_2_1_1"/>
<font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_1_2_1_2"><b>suppose </b></a><a NAME="E1:4_3_1_2_1_2"/><i><font color="Green">E114</font></i>: 
not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">j</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:4_3_1_2_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="abian.html#T1">ABIAN:1</a></i>;<br/><i><font color="Green">E116</font></i>: <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#T51">INT_1:51</a></i>
<br/>.= 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">j</font>

;<br/><a NAME="E5:4_3_1_2_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_1_2_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_3_1_2_1_2_2"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_3_1_2_1_2_2_1"><b>suppose </b></a><a NAME="E1:4_3_1_2_1_2_2_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:4_3_1_2_1_2_2_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E3:4_3_1_2_1_2_2_1"/>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E4:4_3_1_2_1_2_2_1"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:4_3_1_2_1_2_2_1"/>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:4_3_1_2_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/><a NAME="E7:4_3_1_2_1_2_2_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>set </b><font color="Maroon">nm</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">m</font>;<br/><a NAME="E8:4_3_1_2_1_2_2_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>reconsider </b><font color="Maroon">nm</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><b>set </b><font color="Maroon">nm1</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/><a NAME="E10:4_3_1_2_1_2_2_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>reconsider </b><font color="Maroon">nm1</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E12:4_3_1_2_1_2_2_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nm1</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">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>;<br/><a NAME="E13:4_3_1_2_1_2_2_1"/>
<font color="Maroon">nm</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E14:4_3_1_2_1_2_2_1"/>
<font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font>
 <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="4_3_1_2_1_2_2_2"><b>suppose </b></a><a NAME="E1:4_3_1_2_1_2_2_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:4_3_1_2_1_2_2_2"/><b>then </b>
 <a href="binop_2.html#K7">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:4_3_1_2_1_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E4:4_3_1_2_1_2_2_2"/>
<font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E6:4"/>
for <font color="Olive">n</font>, <font color="Olive">k</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> holds <br/><font color="Olive">n</font> <a href="newton.html#K9">choose</a> <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> <a href="newton.html#K9">choose</a> <font color="Olive">k</font>
 ;<br/>


</div>
