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<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="139_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:139_1"/>
( ( <font color="Maroon" title="c2">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="c1">F</font> &amp; <font color="Maroon" title="c3">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="c1">F</font> ) or  not <font color="Maroon" title="c2">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="c1">F</font> or  not <font color="Maroon" title="c3">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="c1">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="139_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:139_1_1"/><i><font color="Green" title="E69">A1</font></i>: 
<font color="Maroon" title="c2">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="c1">F</font>
 <b>and </b><br/><a NAME="E2:139_1_1"/><i><font color="Green" title="E70">A2</font></i>: 
<font color="Maroon" title="c3">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="c1">F</font>
 ;<br/><div class="add"><a NAME="E3:139_1_1"/><i><font color="Green" title="E71">A3</font></i>: 
 <a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="funct_7.html#E1:139_1_1"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="funct_7.html#E2:139_1_1"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="funct_7.html#D12" target="_self" title="FUNCT_7:def.12">Def12</a></i>;<br/><a NAME="E4:139_1_1"/>
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="txt" href="funct_7.html#E1:139_1_1"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E5:139_1_1"/><b>then </b>
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>then </b><i><font color="Green" title="E72">A4</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
<br/>.= 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font>
<b>by </b><i><a class="txt" href="funct_7.html#E2:139_1_1"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
;<br/><a NAME="E7:139_1_1"/><i><font color="Green" title="E73">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></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">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="funct_7.html#E3:139_1_1"><i><font color="Green" title="E71">A3</font></i></a>, <a class="ref" href="funct_7.html#T102" target="_self" title="FUNCT_7:th.102">Th102</a></i>;<br/><a NAME="E8:139_1_1"/>
<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"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="txt" href="funct_7.html#E6:139_1_1"><i><font color="Green" title="E72">A4</font></i></a>, <a class="ref" href="funct_7.html#T102" target="_self" title="FUNCT_7:th.102">Th102</a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></i>;<br/><a NAME="E9:139_1_1"/><b>then </b><i><font color="Green" title="E74">A6</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="txt" href="funct_7.html#E7:139_1_1"><i><font color="Green" title="E73">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:139_1_1"/><i><font color="Green" title="E75">A7</font></i>: 
<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="funct_7.html#E2:139_1_1"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="funct_7.html#T32" target="_self" title="FUNCT_7:th.32">Th32</a></i>;<br/><a NAME="E11:139_1_1"/><b>then </b>
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="funct_7.html#E3:139_1_1"><i><font color="Green" title="E71">A3</font></i></a>, <a class="ref" href="funct_7.html#T104" target="_self" title="FUNCT_7:th.104">Th104</a></i>;<br/><a NAME="E12:139_1_1"/><b>then </b><i><font color="Green" title="E76">A8</font></i>: 
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E13:139_1_1"/><i><font color="Green" title="E77">A9</font></i>: 
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="139_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="139_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:139_1_1_2_1"/>
( <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="139_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:139_1_1_2_1_1"/>
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:139_1_1_2_1_1"/>
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="funct_7.html#E3:139_1_1"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="funct_7.html#E10:139_1_1"><i><font color="Green" title="E75">A7</font></i></a>, <a class="ref" href="funct_7.html#T104" target="_self" title="FUNCT_7:th.104">Th104</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="139_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:139_1_1_2_1_2"/><i><font color="Green" title="E78">A10</font></i>: 
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font>
 ;<br/><div class="add"><a NAME="E2:139_1_1_2_1_2"/><i><font color="Green" title="E79">A11</font></i>: 
 <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="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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="c1">F</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T101" target="_self" title="FUNCT_7:th.101">Th101</a></i>;<br/><span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="funct_7.html#E3:139_1_1"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="funct_7.html#E1:139_1_1_2_1_2"><i><font color="Green" title="E78">A10</font></i></a>, <a class="ref" href="funct_7.html#T34" target="_self" title="FUNCT_7:th.34">Th34</a></i>
<br/>.= 
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="funct_7.html#E1:139_1_1"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="funct_7.html#T33" target="_self" title="FUNCT_7:th.33">Th33</a></i>
;<br/><b>hence </b><a NAME="E4:139_1_1_2_1_2"/>
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="funct_7.html#E1:139_1_1"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="funct_7.html#E2:139_1_1_2_1_2"><i><font color="Green" title="E79">A11</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E14:139_1_1"/><i><font color="Green" title="E78">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="139_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="139_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:139_1_1_3_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="139_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:139_1_1_3_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:139_1_1_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" target="_self" title="FUNCT_7:th.34">Th34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="139_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:139_1_1_3_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:139_1_1_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="funct_7.html#E1:139_1_1"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="funct_7.html#T33" target="_self" title="FUNCT_7:th.33">Th33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><i><font color="Green" title="E79">A13</font></i>: <span class="p1">(<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> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
<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="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="funct_7.html#E12:139_1_1"><i><font color="Green" title="E76">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
<br/>.= 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>
<b>by </b><i><a class="txt" href="funct_7.html#E13:139_1_1"><i><font color="Green" title="E77">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
;<br/><a NAME="E16:139_1_1"/><i><font color="Green" title="E80">A14</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font> <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">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_7.html#T103" target="_self" title="FUNCT_7:th.103">Th103</a></i>;<br/><a NAME="E17:139_1_1"/>
<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"><font color="Maroon" title="c1">F</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="funct_7.html#E3:139_1_1"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="funct_7.html#E14:139_1_1"><i><font color="Green" title="E78">A12</font></i></a>, <a class="txt" href="funct_7.html#E15:139_1_1"><i><font color="Green" title="E79">A13</font></i></a>, <a class="ref" href="funct_7.html#T103" target="_self" title="FUNCT_7:th.103">Th103</a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></i>;<br/><a NAME="E18:139_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</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"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="funct_7.html#E16:139_1_1"><i><font color="Green" title="E80">A14</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E19:139_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="txt" href="funct_7.html#E9:139_1_1"><i><font color="Green" title="E74">A6</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="139_1_2"><b>suppose </b></a><a NAME="E1:139_1_2"/>
(  not <font color="Maroon" title="c2">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="c1">F</font> or  not <font color="Maroon" title="c3">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="c1">F</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:139_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="ref" href="funct_7.html#D12" target="_self" title="FUNCT_7:def.12">Def12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
