<?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="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="exchsort.html#V2" title="EXCHSORT:attr.2">-based</a>   <a href="exchsort.html#M2" title="EXCHSORT:mode.2">arr_computation</a> of <font color="Olive" title="b1">R</font>  st <font color="Olive" title="b1">R</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="exchsort.html#NM3" title="EXCHSORT:NM.3">array</a> of <font color="Maroon" title="c1">O</font> holds <br/><font color="Olive" title="b2">C</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </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="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="exchsort.html#V2" title="EXCHSORT:attr.2">-based</a>   <a href="exchsort.html#M2" title="EXCHSORT:mode.2">arr_computation</a> of <font color="Maroon" title="c2">R</font>  st <font color="Maroon" title="c2">R</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="exchsort.html#NM3" title="EXCHSORT:NM.3">array</a> of <font color="Maroon" title="c1">O</font> holds <br/><font color="Olive" title="b1">C</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">C</font> be   <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="exchsort.html#V2" title="EXCHSORT:attr.2">-based</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"> ( <font color="Maroon" title="c2">R</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="exchsort.html#NM3" title="EXCHSORT:NM.3">array</a> of <font color="Maroon" title="c1">O</font> implies <font color="Maroon" title="c3">C</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:127"/>
<font color="Maroon" title="c2">R</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <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"> <font color="Maroon" title="c3">C</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">R</font> = <font color="Maroon" title="c2">R</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="exchsort.html#NM3" title="EXCHSORT:NM.3">array</a> of <font color="Maroon" title="c1">O</font> ;<br/>
<a NAME="E3:127"/><span class="lab"><font color="Green" title="E71">A0</font></span>: 
<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="c4">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/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="exchsort.html#NM3" title="EXCHSORT:NM.3">array</a> of <font color="Maroon" title="c1">O</font>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> $1</span>)</span>;<br/>
<a NAME="E4:127"/>
 <a href="exchsort.html#K1" title="EXCHSORT:func.1">base-</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="exchsort.html#T24" target="_self" title="EXCHSORT:th.24">TT0</a></span>;<br/>
<a NAME="E5:127"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A1</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127"><span class="lab"><font color="Green" title="E71">A0</font></span></a></span>;<br/>
<a NAME="E6:127"/><span class="lab"><font color="Green" title="E73">A2</font></span>: 
 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  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <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> &amp; <font color="Maroon">H<sub>1</sub></font>(<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="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  holds <br/><font color="Maroon">H<sub>1</sub></font>(<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="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">a</font></span>)</span>) <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<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>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</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"> (  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">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> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  implies <font color="Maroon">H<sub>1</sub></font>(<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="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">a</font></span>)</span>) <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:127_1"/><span class="lab"><font color="Green" title="E74">Z1</font></span>: 
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">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> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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">H<sub>1</sub></font>(<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="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">a</font></span>)</span>) <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>)</span><br/>

<a NAME="E2:127_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a></span>;<br/>
<a NAME="E3:127_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1"><span class="lab"><font color="Green" title="E74">Z1</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">R</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="c7">x</font>, <font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:127_1"/><span class="lab"><font color="Green" title="E75">A3</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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="c6">R</font> &amp; <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">R</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"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">a</font></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="c6">R</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1"><span class="lab"><font color="Green" title="E74">Z1</font></span></a>, <a class="ref" href="exchsort.html#D14" target="_self" title="EXCHSORT:def.14">COMP</a></span>;<br/>
<a NAME="E6:127_1"/>
 <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <font color="Maroon" title="c6">R</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1"><span class="lab"><font color="Green" title="E74">Z1</font></span></a>, <a class="txt" href="#E5:127_1"><span class="lab"><font color="Green" title="E75">A3</font></span></a></span>;<br/>
<a NAME="E7:127_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<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="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">a</font></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:127_1"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="ref" href="exchsort.html#T73" target="_self" title="EXCHSORT:th.73">MAIN3</a></span>;<br/>
<a NAME="E8:127_1"/><span class="kw">then </span>
( <font color="Maroon">H<sub>1</sub></font>(<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="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">a</font></span>)</span>) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<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="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">a</font></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:127_1"/>
<font color="Maroon">H<sub>1</sub></font>(<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="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">a</font></span>)</span>) <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</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>
<span class="kw">thus </span><a NAME="E7:127"/>
<font color="Maroon" title="c3">C</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">from</span> <span class="lab"><a class="ref" href="exchsort.html#S1" target="_self" title="EXCHSORT:sch.1">EXCHSORT:sch 1</a>(<a class="txt" href="#E5:127"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="txt" href="#E6:127"><span class="lab"><font color="Green" title="E73">A2</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"> verum</span><br/>


</div>
