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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">set </span><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/>
<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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">ORDINAL1:24</a></span>;<br/>
<a NAME="E3:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T10" title="ARYTM_3:th.10">ARYTM_3:10</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:17_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="numbers.html#E16"><span class="lab"><font color="Green" title="E12">Lm11</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><span class="kw">suppose </span></a><a NAME="E1:17_1_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">take </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">thus </span><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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="ordinal3.html#T78" title="ORDINAL3:th.78">ORDINAL3:78</a></span>
<br/>.= 
2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c4">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2"><span class="kw">suppose </span></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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"/><span class="kw">then </span>
<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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T78" title="ORDINAL3:th.78">ORDINAL3:78</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A4</font></span>: <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T54" title="ORDINAL3:th.54">ORDINAL3:54</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T54" title="ORDINAL3:th.54">ORDINAL3:54</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a></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" 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
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T33" title="ORDINAL3:th.33">ORDINAL3:33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">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
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T54" title="ORDINAL3:th.54">ORDINAL3:54</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">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
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T58" title="ORDINAL3:th.58">ORDINAL3:58</a></span>
;<br/><a NAME="E4:17_1_2"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
1 <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T14" title="ARYTM_3:th.14">ARYTM_3:14</a></span>;<br/><a NAME="E5:17_1_2"/>
( 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 &amp; 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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="arytm_3.html#D3" title="ARYTM_3:def.3">ARYTM_3:def 3</a></span>;<br/><a NAME="E6:17_1_2"/><span class="kw">then </span>
2 <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_2"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="arytm_3.html#T16" title="ARYTM_3:th.16">ARYTM_3:16</a></span>;<br/><span class="kw">hence </span><a NAME="E7:17_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_2"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="arytm_3.html#T13" title="ARYTM_3:th.13">ARYTM_3:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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