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

<span class="kw">set </span><font color="Maroon" title="c6">X</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">R</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">i</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">R</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">s</font> =  <a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c9">h</font> = <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">Y</font> = <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">x</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">Z1</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">Z2</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">g</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>;<br/>
<a NAME="E1:114_1_1"/><span class="lab"><font color="Green" title="E65">A0</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c5">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="ref" href="relat_1.html#T111" title="RELAT_1:th.111">RELAT_1:111</a></span>;<br/>
<a NAME="E2:114_1_1"/><span class="lab"><font color="Green" title="E66">A1</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span></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="funct_3.html#T67" title="FUNCT_3:th.67">FUNCT_3:67</a></span>;<br/>
<a NAME="E3:114_1_1"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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"><span class="p3">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></span>;<br/>
<a NAME="E4:114_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> (<span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><span class="p4"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T17" title="FUNCT_4:th.17">FUNCT_4:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:114_1_1"/>
<span class="p1">(<span class="default">(<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c5">z</font> is  <a href="relat_1.html#V1" title="RELAT_1:attr.1">Relation-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_1"><span class="lab"><font color="Green" title="E65">A0</font></span></a>, <a class="txt" href="#E2:114_1_1"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E3:114_1_1"><span class="lab"><font color="Green" title="E67">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>
