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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> holds <br/><span class="p1">(<span class="default">(<font color="Olive" title="b2">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="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="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="Olive" title="b2">R</font></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> holds <br/><span class="p1">(<span class="default">(<font color="Olive" title="b1">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="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="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="Olive" title="b1">R</font></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> implies <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><br/>







<span class="kw">assume </span><a NAME="E1:115"/><span class="lab"><font color="Green" title="E66">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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><br/>

<a NAME="E2:115"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">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> &amp; <font color="Maroon" title="c4">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x</font> <a href="orders_2.html#NR4" title="ORDERS_2:NR.4">&gt;</a> <font color="Maroon" title="c4">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="exchsort.html#T46" target="_self" title="EXCHSORT:th.46">TW0</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">T</font> = <font color="Maroon" title="c4">R</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="exchsort.html#NM3" title="EXCHSORT:NM.3">array</a> of <font color="Maroon" title="c3">O</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E66">Z0</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c7">q</font> = <font color="Maroon" title="c2">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E66">Z0</font></span></a>, <a class="ref" href="exchsort.html#T46" target="_self" title="EXCHSORT:th.46">TW0</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">j</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="c9">k</font> = (<font color="Maroon" title="c5">T</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">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="c11">t</font> =  <a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<font color="Maroon" title="c5">T</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c12">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">set </span><font color="Maroon" title="c13">w</font> =  <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <font color="Maroon" title="c4">R</font>;<br/>
<a NAME="E5:115"/><span class="lab"><font color="Green" title="E68">A2</font></span>: 
 <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="c5">T</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</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="c5">T</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/>
<span class="kw">thus </span><a NAME="E6:115"/>
<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>  <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <font color="Maroon" title="c4">R</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="exchsort.html#K6" title="EXCHSORT:func.6">inversions</a> <font color="Maroon" title="c4">R</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</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">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Olive">b<sub>1</sub></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> )</span><br/><span class="kw">let </span><font color="Maroon" title="c15">t</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"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</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> )</span><br/>



<span class="kw">assume </span><a NAME="E1:115_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>
 ; <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="c14">z</font>,<font color="Maroon" title="c15">t</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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">a</font>, <font color="Maroon" title="c17">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:115_1"/><span class="lab"><font color="Green" title="E69">A0</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c17">b</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> <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; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c17">b</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="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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c17">b</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="exchsort.html#T5" target="_self" title="EXCHSORT:th.5">Lem3</a></span>;<br/>
<a NAME="E4:115_1"/><span class="lab"><font color="Green" title="E70">A3</font></span>: 
( <font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">b</font> &amp; <font color="Maroon" title="c16">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="c5">T</font> &amp; <font color="Maroon" title="c17">b</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="c5">T</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E5:115"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="exchsort.html#T46" target="_self" title="EXCHSORT:th.46">TW0</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">a</font> = <font color="Maroon" title="c16">a</font>, <font color="Maroon" title="c19">b</font> = <font color="Maroon" title="c17">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E5:115"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="exchsort.html#T46" target="_self" title="EXCHSORT:th.46">TW0</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_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="E1:115_1_1"/>
( ( <font color="Maroon" title="c18">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c18">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q</font> ) or ( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> ) or ( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> ) or ( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">q</font> ) or ( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> ) or ( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">b</font> ) or ( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">b</font> ) or ( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">a</font> &amp; <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E4:115_1"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="ref" href="exchsort.html#T2" target="_self" title="EXCHSORT:th.2">Eight</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:115_1_1_1"/><span class="lab"><font color="Green" title="E71">C1</font></span>: 
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c18">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q</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="c14">z</font>,<font color="Maroon" title="c15">t</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></span><br/><div class="add"><a NAME="E2:115_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">a</font>,<font color="Maroon" title="c19">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="exchsort.html#T67" target="_self" title="EXCHSORT:th.67">Case1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:115_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E1:115_1_1_1"><span class="lab"><font color="Green" title="E71">C1</font></span></a>, <a class="ref" href="exchsort.html#T52" target="_self" title="EXCHSORT:th.52">TC1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:115_1_1_2"/><span class="lab"><font color="Green" title="E71">C2</font></span>: 
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</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="c14">z</font>,<font color="Maroon" title="c15">t</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></span><br/><div class="add"><a NAME="E2:115_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">a</font>,<font color="Maroon" title="c7">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:115_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E66">Z0</font></span></a>, <a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E1:115_1_1_2"><span class="lab"><font color="Green" title="E71">C2</font></span></a>, <a class="ref" href="exchsort.html#T53" target="_self" title="EXCHSORT:th.53">TC2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:115_1_1_3"/><span class="lab"><font color="Green" title="E71">C3</font></span>: 
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</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="c14">z</font>,<font color="Maroon" title="c15">t</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></span><br/><div class="add"><a NAME="E2:115_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">a</font>,<font color="Maroon" title="c6">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:115_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E66">Z0</font></span></a>, <a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E1:115_1_1_3"><span class="lab"><font color="Green" title="E71">C3</font></span></a>, <a class="ref" href="exchsort.html#T54" target="_self" title="EXCHSORT:th.54">TC3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:115_1_1_4"/><span class="lab"><font color="Green" title="E71">C4</font></span>: 
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">q</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="c14">z</font>,<font color="Maroon" title="c15">t</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></span><br/><div class="add"><a NAME="E2:115_1_1_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">a</font>,<font color="Maroon" title="c19">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115_1"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:115_1_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E66">Z0</font></span></a>, <a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E1:115_1_1_4"><span class="lab"><font color="Green" title="E71">C4</font></span></a>, <a class="ref" href="exchsort.html#T55" target="_self" title="EXCHSORT:th.55">TC4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:115_1_1_5"/>
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</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="c14">z</font>,<font color="Maroon" title="c15">t</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></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:115_1_1_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E66">Z0</font></span></a>, <a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="exchsort.html#T66" target="_self" title="EXCHSORT:th.66">Case5</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_6"><span class="kw">suppose </span></a><a NAME="E1:115_1_1_6"/><span class="lab"><font color="Green" title="E71">C6</font></span>: 
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">b</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="c14">z</font>,<font color="Maroon" title="c15">t</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></span><br/><div class="add"><a NAME="E2:115_1_1_6"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c19">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:115_1_1_6"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E66">Z0</font></span></a>, <a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E1:115_1_1_6"><span class="lab"><font color="Green" title="E71">C6</font></span></a>, <a class="ref" href="exchsort.html#T57" target="_self" title="EXCHSORT:th.57">TC6</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_7"><span class="kw">suppose </span></a><a NAME="E1:115_1_1_7"/><span class="lab"><font color="Green" title="E71">C7</font></span>: 
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">b</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="c14">z</font>,<font color="Maroon" title="c15">t</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></span><br/><div class="add"><a NAME="E2:115_1_1_7"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c19">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:115_1_1_7"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E66">Z0</font></span></a>, <a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E1:115_1_1_7"><span class="lab"><font color="Green" title="E71">C7</font></span></a>, <a class="ref" href="exchsort.html#T58" target="_self" title="EXCHSORT:th.58">TC7</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_8"><span class="kw">suppose </span></a><a NAME="E1:115_1_1_8"/><span class="lab"><font color="Green" title="E71">C8</font></span>: 
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">a</font> &amp; <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</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="c14">z</font>,<font color="Maroon" title="c15">t</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></span><br/><div class="add"><a NAME="E2:115_1_1_8"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">a</font>,<font color="Maroon" title="c19">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E4:115_1"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:115_1_1_8"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c15">t</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E66">Z0</font></span></a>, <a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E1:115_1_1_8"><span class="lab"><font color="Green" title="E71">C8</font></span></a>, <a class="ref" href="exchsort.html#T56" target="_self" title="EXCHSORT:th.56">TC8</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:115_2"/>
<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> <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 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> <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><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:115_2"/><span class="lab"><font color="Green" title="E69">A0</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</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> <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; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</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="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> &amp; <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#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="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="exchsort.html#T5" target="_self" title="EXCHSORT:th.5">Lem3</a></span>;<br/><a NAME="E4:115_2"/><span class="lab"><font color="Green" title="E70">A5</font></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="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="exchsort.html#T66" target="_self" title="EXCHSORT:th.66">Case5</a></span>;<br/><a NAME="E5:115_2"/><span class="lab"><font color="Green" title="E71">A6</font></span>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> &amp; <font color="Maroon" title="c14">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="c5">T</font> &amp; <font color="Maroon" title="c15">b</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="c5">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_2"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E5:115"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="exchsort.html#T46" target="_self" title="EXCHSORT:th.46">TW0</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">a</font> = <font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c17">b</font> = <font color="Maroon" title="c15">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_2"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E5:115"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="exchsort.html#T46" target="_self" title="EXCHSORT:th.46">TW0</a></span>;<br/><a NAME="E7:115_2"/>
( <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E3:115_2"><span class="lab"><font color="Green" title="E69">A0</font></span></a>, <a class="txt" href="#E4:115_2"><span class="lab"><font color="Green" title="E70">A5</font></span></a>, <a class="txt" href="#E5:115_2"><span class="lab"><font color="Green" title="E71">A6</font></span></a>, <a class="txt" href="exchsort.html#E84"><span class="lab"><font color="Green" title="E63">O2O</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:115_2"/>
<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> <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="txt" href="#E3:115_2"><span class="lab"><font color="Green" title="E69">A0</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:115"/>
 not <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="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E1:115"><span class="lab"><font color="Green" title="E66">Z0</font></span></a>, <a class="ref" href="exchsort.html#T51" target="_self" title="EXCHSORT:th.51">TC5</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/>


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