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

<span class="kw">let </span><font color="Maroon" title="c1">O</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being   <a href="exchsort.html#NM3" title="EXCHSORT:NM.3">array</a> of <font color="Maroon" title="c1">O</font><br/>  for <font color="Olive" title="b2">C</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="exchsort.html#M2" title="EXCHSORT:mode.2">arr_computation</a> of <font color="Olive" title="b1">R</font> holds <br/> (  <a href="exchsort.html#K4" title="EXCHSORT:func.4">last</a> <font color="Olive" title="b2">C</font> is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Olive" title="b1">R</font> &amp; (  for <font color="Olive" title="b3">a</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">C</font> holds <br/><font color="Olive" title="b2">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">a</font> is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Olive" title="b1">R</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="exchsort.html#NM3" title="EXCHSORT:NM.3">array</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="exchsort.html#M2" title="EXCHSORT:mode.2">arr_computation</a> of <font color="Maroon" title="c2">R</font> holds <br/> (  <a href="exchsort.html#K4" title="EXCHSORT:func.4">last</a> <font color="Olive" title="b1">C</font> is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Maroon" title="c2">R</font> &amp; (  for <font color="Olive" title="b2">a</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">C</font> holds <br/><font color="Olive" title="b1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">a</font> is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Maroon" title="c2">R</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">C</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="exchsort.html#M2" title="EXCHSORT:mode.2">arr_computation</a> of <font color="Maroon" title="c2">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="exchsort.html#K4" title="EXCHSORT:func.4">last</a> <font color="Maroon" title="c3">C</font> is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Maroon" title="c2">R</font> &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">C</font> holds <br/><font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Maroon" title="c2">R</font> ) )</span><br/>





<span class="kw">consider </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E2:129"/><span class="lab"><font color="Green" title="E72">A0</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="exchsort.html#D1" target="_self" title="EXCHSORT:def.1">SEG</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">n</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E4:129"/><span class="lab"><font color="Green" title="E73">A1</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E72">A0</font></span></a>, <a class="ref" href="exchsort.html#T7" target="_self" title="EXCHSORT:th.7">Th009</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>9</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( <font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">C</font> implies <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1</span>)</span> is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Maroon" title="c2">R</font> );<br/>
<a NAME="E5:129"/><span class="lab"><font color="Green" title="E74">B1</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></span>;<br/>
<a NAME="E6:129"/><span class="kw">then </span>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E72">A0</font></span></a>, <a class="txt" href="#E4:129"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:129"/><span class="lab"><font color="Green" title="E75">B2</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/>
<span class="lab"><font color="Green" title="E76">B5</font></span>: <font color="Maroon" title="c4">a</font> = 
<font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:129"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E8:129"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="ref" href="nat_1.html#T38" title="NAT_1:th.38">NAT_1:38</a></span>
<br/>.= 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c7">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T28" title="ORDINAL2:th.28">ORDINAL2:28</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E77">B4</font></span>: <font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c7">m</font> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T2" title="ORDINAL2:th.2">ORDINAL2:2</a></span>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E72">A0</font></span></a>, <a class="ref" href="exchsort.html#T6" target="_self" title="EXCHSORT:th.6">Th008</a></span>
;<br/>
<a NAME="E11:129"/>
<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="exchsort.html#K1" title="EXCHSORT:func.1">base-</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="exchsort.html#D14" target="_self" title="EXCHSORT:def.14">COMP</a></span>;<br/>
<a NAME="E12:129"/><span class="kw">then </span>
<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E72">A0</font></span></a>, <a class="ref" href="exchsort.html#T23" target="_self" title="EXCHSORT:th.23">Th011</a></span>;<br/>
<a NAME="E13:129"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A2</font></span>: 
<font color="Maroon">S<sub>9</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:129"><span class="lab"><font color="Green" title="E74">B1</font></span></a>, <a class="ref" href="exchsort.html#T38" target="_self" title="EXCHSORT:th.38">ThRef</a></span>;<br/>
<a NAME="E14:129"/><span class="lab"><font color="Green" title="E79">A3</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon">S<sub>9</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>9</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:129_3"/><span class="lab"><font color="Green" title="E80">C1</font></span>: 
( <font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c8">k</font>] &amp; <font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Maroon" title="c2">R</font></span><br/>

<a NAME="E2:129_3"/>
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T38" title="NAT_1:th.38">NAT_1:38</a></span>;<br/>
<a NAME="E3:129_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">C2</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T28" title="ORDINAL2:th.28">ORDINAL2:28</a></span>;<br/>
<a NAME="E4:129_3"/><span class="kw">then </span>
( <font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E72">A0</font></span></a>, <a class="txt" href="#E1:129_3"><span class="lab"><font color="Green" title="E80">C1</font></span></a>, <a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a></span>;<br/>
<a NAME="E5:129_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">C3</font></span>: 
( <font color="Maroon" title="c5">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="ordinal3.html#T24" title="ORDINAL3:th.24">ORDINAL3:24</a></span>;<br/>
<a NAME="E6:129_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E72">A0</font></span></a>, <a class="ref" href="ordinal6.html#T5" title="ORDINAL6:th.5">ORDINAL6:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">Q</font> being   <a href="exchsort.html#NM3" title="EXCHSORT:NM.3">array</a> of <font color="Maroon" title="c1">O</font>, <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:129_3"/><span class="lab"><font color="Green" title="E83">C4</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <font color="Maroon" title="c9">Q</font> &amp; <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Q</font> &amp; <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c9">Q</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129_3"><span class="lab"><font color="Green" title="E80">C1</font></span></a>, <a class="txt" href="#E3:129_3"><span class="lab"><font color="Green" title="E81">C2</font></span></a>, <a class="ref" href="exchsort.html#D14" target="_self" title="EXCHSORT:def.14">COMP</a></span>;<br/>
<a NAME="E9:129_3"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">Q</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">Q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:129_3"><span class="lab"><font color="Green" title="E83">C4</font></span></a>, <a class="ref" href="exchsort.html#T46" target="_self" title="EXCHSORT:th.46">TW0</a></span>;<br/>
<a NAME="E10:129_3"/><span class="kw">then </span>
 <a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c9">Q</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>) is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Maroon" title="c9">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="exchsort.html#T43" target="_self" title="EXCHSORT:th.43">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:129_3"/>
<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129_3"><span class="lab"><font color="Green" title="E80">C1</font></span></a>, <a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E72">A0</font></span></a>, <a class="txt" href="#E5:129_3"><span class="lab"><font color="Green" title="E82">C3</font></span></a>, <a class="txt" href="#E8:129_3"><span class="lab"><font color="Green" title="E83">C4</font></span></a>, <a class="ref" href="exchsort.html#T40" target="_self" title="EXCHSORT:th.40">ThTr</a>, <a class="ref" href="ordinal6.html#T5" title="ORDINAL6:th.5">ORDINAL6:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:129"/><span class="lab"><font color="Green" title="E80">A4</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon">S<sub>9</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E13:129"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="txt" href="#E14:129"><span class="lab"><font color="Green" title="E79">A3</font></span></a>);<br/></span>
<a NAME="E16:129"/>
( <font color="Maroon" title="c5">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:129"><span class="lab"><font color="Green" title="E76">B5</font></span></a>, <a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a>, <a class="ref" href="ordinal3.html#T24" title="ORDINAL3:th.24">ORDINAL3:24</a></span>;<br/>
<a NAME="E17:129"/><span class="kw">then </span>
( <font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c7">m</font>] &amp; <font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E72">A0</font></span></a>, <a class="txt" href="#E15:129"><span class="lab"><font color="Green" title="E80">A4</font></span></a>, <a class="ref" href="ordinal6.html#T5" title="ORDINAL6:th.5">ORDINAL6:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:129"/>
 <a href="exchsort.html#K4" title="EXCHSORT:func.4">last</a> <font color="Maroon" title="c3">C</font> is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E72">A0</font></span></a>, <a class="txt" href="#E10:129"><span class="lab"><font color="Green" title="E77">B4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">C</font> holds <br/><font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Maroon" title="c2">R</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">c</font> be   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">C</font> implies <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">c</font> is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Maroon" title="c2">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E19:129"/><span class="lab"><font color="Green" title="E81">A7</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">c</font> is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Maroon" title="c2">R</font></span><br/>

<a NAME="E20:129"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A5</font></span>: 
( <font color="Maroon" title="c5">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E72">A0</font></span></a>, <a class="ref" href="ordinal6.html#T5" title="ORDINAL6:th.5">ORDINAL6:5</a></span>;<br/>
<a NAME="E21:129"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c5">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>;<br/>
<a NAME="E22:129"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:129"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E19:129"><span class="lab"><font color="Green" title="E81">A7</font></span></a>, <a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E72">A0</font></span></a>, <a class="ref" href="ordinal3.html#T22" title="ORDINAL3:th.22">ORDINAL3:22</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">k</font> = <font color="Maroon" title="c8">c</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c5">b</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E24:129"/>
<font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c9">k</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:129"><span class="lab"><font color="Green" title="E80">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:129"/>
<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">c</font> is    <a href="exchsort.html#M1" title="EXCHSORT:mode.1">permutation</a> of <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:129"><span class="lab"><font color="Green" title="E81">A7</font></span></a>, <a class="txt" href="#E20:129"><span class="lab"><font color="Green" title="E82">A5</font></span></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
