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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">O</font> being   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><br/>  for <font color="Olive" title="b2">R</font> being   <a href="exchsort.html#NM3" title="EXCHSORT:NM.3">array</a> of <font color="Olive" title="b1">O</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">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="Olive" title="b2">R</font> &amp;  <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <font color="Olive" title="b2">R</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  holds <br/> <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> <span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Olive" title="b2">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Olive" title="b2">R</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="c3">O</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">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="Olive" title="b1">R</font> &amp;  <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <font color="Olive" title="b1">R</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  holds <br/> <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> <span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Olive" title="b1">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Olive" title="b1">R</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">R</font> be   <a href="exchsort.html#NM3" title="EXCHSORT:NM.3">array</a> of <font color="Maroon" title="c3">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">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="c4">R</font> &amp;  <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <font color="Maroon" title="c4">R</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  implies  <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> <span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c4">R</font></span>)</span> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">s</font> =  <a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">h</font> = (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">ws</font> =  <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:117"/><span class="lab"><font color="Green" title="E67">Z0</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">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="c4">R</font> &amp;  <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <font color="Maroon" title="c4">R</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">  <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> <span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c4">R</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">w</font> =  <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <font color="Maroon" title="c4">R</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E3:117"/>
<span class="p1">(<span class="default">(<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <font color="Maroon" title="c4">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E67">Z0</font></span></a>, <a class="ref" href="exchsort.html#T72" target="_self" title="EXCHSORT:th.72">MAIN1</a></span>;<br/>
<a NAME="E4:117"/><span class="kw">then </span>
<span class="p1">(<span class="default">(<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">w</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>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">hws</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E6:117"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">hws</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E67">Z0</font></span></a>, <a class="ref" href="exchsort.html#T72" target="_self" title="EXCHSORT:th.72">MAIN1</a>, <a class="ref" href="trees_1.html#T6" title="TREES_1:th.6">TREES_1:6</a></span>;<br/>
<a NAME="E7:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A3</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">hws</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E8:117"/><span class="lab"><font color="Green" title="E69">A1</font></span>: 
( <font color="Maroon" title="c1">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="c4">R</font> &amp; <font color="Maroon" title="c2">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="c4">R</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E67">Z0</font></span></a>, <a class="ref" href="exchsort.html#T46" target="_self" title="EXCHSORT:th.46">TW0</a></span>;<br/>
<a NAME="E9:117"/><span class="lab"><font color="Green" title="E70">A2</font></span>: 
 not <font color="Maroon" title="c4">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E67">Z0</font></span></a></span>;<br/>
<a NAME="E10:117"/>
 <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span>,<span class="p1">(<span class="default">(<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c10">hw</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</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="c10">hw</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c10">hw</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c10">hw</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E1:117_1"/>
<font color="Maroon" title="c10">hw</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:117"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="txt" href="#E9:117"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="exchsort.html#T71" target="_self" title="EXCHSORT:th.71">one2one</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c10">hw</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c10">hw</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> )</span><br/>

<a NAME="E2:117_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T99" title="FUNCT_7:th.99">FUNCT_7:99</a></span>;<br/>
<a NAME="E3:117_1"/><span class="kw">then </span>
 <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">R</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="exchsort.html#T47" target="_self" title="EXCHSORT:th.47">TW1</a></span>;<br/>
<a NAME="E4:117_1"/><span class="kw">then </span>
 <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:117"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="txt" href="#E9:117"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="exchsort.html#T61" target="_self" title="EXCHSORT:th.61">FF2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:117_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">hw</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c10">hw</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E6:117_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c10">hw</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T115" title="RELAT_1:th.115">RELAT_1:115</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">hence </span><a NAME="E11:117"/>
 <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> <span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c4">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:117"><span class="lab"><font color="Green" title="E68">A3</font></span></a>, <a class="ref" href="card_1.html#T5" title="CARD_1:th.5">CARD_1:5</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>
