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

<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">k</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c2">k</font> implies  ex <font color="Olive" title="b1">k</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b1">k</font> )</span><br/>

<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c2">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b1">k</font></span><br/>

<a NAME="E2:17"/>
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2
 
<b>by </b><i><a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">ORDINAL1:24</a></i>;<br/>
<a NAME="E3:17"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2
 
<b>by </b><i><a class="ref" href="arytm_3.html#T10" title="ARYTM_3:th.10">ARYTM_3:10</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">id2</font> = <font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1"/>
( <font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="numbers.html#E3:17"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="numbers.html#E16"><i><font color="Green" title="E12">Lm11</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><b>suppose </b></a><a NAME="E1:17_1_1"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b1">k</font></span><br/><div class="add"><b>take </b><font color="Maroon" title="c4">k</font> = <font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c4">k</font></span><br/><b>thus </b><font color="Maroon" title="c1">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="numbers.html#E1:17_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="ordinal3.html#T78" title="ORDINAL3:th.78">ORDINAL3:78</a></i>
<br/>.= 
2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c4">k</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2"><b>suppose </b></a><a NAME="E1:17_1_2"/>
<font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b1">k</font></span><br/><div class="add"><a NAME="E2:17_1_2"/><b>then </b>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1
 <b>by </b><i><a class="ref" href="ordinal3.html#T78" title="ORDINAL3:th.78">ORDINAL3:78</a></i>;<br/><b>then </b><i><font color="Green" title="E16">A4</font></i>: <font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1</span>)</span></span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T54" title="ORDINAL3:th.54">ORDINAL3:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1</span>)</span></span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span></span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T54" title="ORDINAL3:th.54">ORDINAL3:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1</span>)</span></span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span></span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</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" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1</span>)</span></span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span></span>)</span></span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1
<b>by </b><i><a class="ref" href="ordinal3.html#T33" title="ORDINAL3:th.33">ORDINAL3:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1</span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> </span>)</span></span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1
<b>by </b><i><a class="ref" href="ordinal3.html#T54" title="ORDINAL3:th.54">ORDINAL3:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1</span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> </span>)</span></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1
<b>by </b><i><a class="ref" href="ordinal3.html#T58" title="ORDINAL3:th.58">ORDINAL3:58</a></i>
;<br/><a NAME="E4:17_1_2"/><i><font color="Green" title="E17">A5</font></i>: 
2 <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> 2</span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2</span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1</span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> </span>)</span></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2
 <b>by </b><i><a class="ref" href="arytm_3.html#D3" title="ARYTM_3:def.3">ARYTM_3:def 3</a></i>;<br/><a NAME="E5:17_1_2"/>
2 <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c1">i</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c1">i</font>
 <b>by </b><i><a class="txt" href="numbers.html#E1:17"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="arytm_3.html#D3" title="ARYTM_3:def.3">ARYTM_3:def 3</a></i>;<br/><a NAME="E6:17_1_2"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
2 <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> 1
 <b>by </b><i><a class="txt" href="numbers.html#E3:17_1_2"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="numbers.html#E4:17_1_2"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="arytm_3.html#T16" title="ARYTM_3:th.16">ARYTM_3:16</a></i>;<br/><a NAME="E7:17_1_2"/>
1 <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> 2
 <b>by </b><i><a class="ref" href="arytm_3.html#T14" title="ARYTM_3:th.14">ARYTM_3:14</a></i>;<br/><b>hence </b><a NAME="E8:17_1_2"/>
 ex <font color="Olive" title="b1">k</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="txt" href="numbers.html#E6:17_1_2"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="arytm_3.html#T13" title="ARYTM_3:th.13">ARYTM_3:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
