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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>set </b><font color="Maroon" title="c5">p1</font> = <font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">p2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>;<br/>
<a NAME="E1:1"/><i><font color="Green" title="E2">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/>
<a NAME="E2:1"/><i><font color="Green" title="E3">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/>
<a NAME="E3:1"/>
( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
;<br/>
<a NAME="E4:1"/><b>then </b><i><font color="Green" title="E4">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1"/>
( <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1"/>
<font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="funct_2.html#T55" title="FUNCT_2:th.55">FUNCT_2:55</a>, <a class="ref" href="partfun1.html#T58" title="PARTFUN1:th.58">PARTFUN1:58</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2"><b>suppose </b></a><a NAME="E1:1_1_2"/><i><font color="Green" title="E5">A4</font></i>: 
not <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:1_1_2"/><i><font color="Green" title="E6">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><a NAME="E3:1_1_2"/><b>then </b><i><font color="Green" title="E7">A6</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E4:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>thus </b><a NAME="E4:1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_2_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_1"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> = 
<font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T37" title="FUNCT_7:th.37">FUNCT_7:37</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="txt" href="sfmastr3.html#E1:1_1_2_1_1_1"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="funct_7.html#T37" title="FUNCT_7:th.37">FUNCT_7:37</a></i>
;<br/><b>hence </b><a NAME="E3:1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_2"/><i><font color="Green" title="E8">A8</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E9">A9</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font>
<b>by </b><i><a class="txt" href="sfmastr3.html#E4:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>
;<br/><a NAME="E3:1_1_2_1_1_2"/><i><font color="Green" title="E10">A10</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><div><i><font color="Green" title="E11">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_2_1_1_2_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E2:1_1_2_1_1_2_2"/><i><font color="Green" title="E12">A12</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> )
 ;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font>
<b>by </b><i><a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2_2"><i><font color="Green" title="E12">A12</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font>
<b>by </b><i><a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2_2"><i><font color="Green" title="E12">A12</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
;<br/></div><b>end;</b></div><div><i><font color="Green" title="E12">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">pz</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:1_1_2_1_1_2_3_1"/>
<font color="Maroon" title="c7">pz</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:1_1_2_1_1_2_3_1"/><i><font color="Green" title="E13">A14</font></i>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">pz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:1_1_2_1_1_2_3_1"/><i><font color="Green" title="E14">A15</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1_1_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2_3_1"><i><font color="Green" title="E13">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_2_1_1_2_3_1_1"/>
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> or <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> or ( <font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon" title="c7">pz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E4:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1_1_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2_3_1"><i><font color="Green" title="E13">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon" title="c7">pz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2_3_1"><i><font color="Green" title="E13">A14</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_2_3_1_1_3"/>
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_2_1_1_2_3_1_1_3"/>
<font color="Maroon" title="c7">pz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2_3_1"><i><font color="Green" title="E13">A14</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2_3_1"><i><font color="Green" title="E14">A15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:1_1_2_1_1_2_3"/>
<font color="Maroon" title="c7">pz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:1_1_2_1_1_2_3"/><i><font color="Green" title="E13">A16</font></i>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c7">pz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> )
 <b>by </b><i><a class="txt" href="sfmastr3.html#E2:1_1_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_2_1_1_2_3_2"/>
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> or <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> or ( <font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_3_2_1"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_2_3_2_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_2_1_1_2_3_2_1"/>
<font color="Maroon" title="c7">pz</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2_3"><i><font color="Green" title="E13">A16</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_3_2_2"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_2_3_2_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_2_1_1_2_3_2_2"/>
<font color="Maroon" title="c7">pz</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2_3"><i><font color="Green" title="E13">A16</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_3_2_3"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_2_3_2_3"/>
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_2_1_1_2_3_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2_3"><i><font color="Green" title="E13">A16</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_2_1_1_2_3_2_3"/>
<font color="Maroon" title="c7">pz</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2_3"><i><font color="Green" title="E13">A16</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E6:1_1_2_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">p2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="sfmastr3.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/><a NAME="E8:1_1_2_1_1_2"/>
<font color="Maroon" title="c7">p2</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_1_1_2_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_1_1_2_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">z1</font>, <font color="Maroon" title="c9">z2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_2_1_1_2_4_1"/><i><font color="Green" title="E13">A17</font></i>: 
<font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E2:1_1_2_1_1_2_4_1"/><i><font color="Green" title="E14">A18</font></i>: 
<font color="Maroon" title="c9">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E3:1_1_2_1_1_2_4_1"/><i><font color="Green" title="E15">A19</font></i>: 
<font color="Maroon" title="c7">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z2</font>
 <b>and </b><br/><a NAME="E4:1_1_2_1_1_2_4_1"/><i><font color="Green" title="E16">A20</font></i>: 
<font color="Maroon" title="c8">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">z2</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_2_1_1_2_4_1_1"/>
( ( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> ) or ( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> ) or ( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> ) or ( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> ) or ( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> ) or ( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> ) or ( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c8">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_2_4_1_1_1"/>
( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_2_1_1_2_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:1_1_2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2_1_1_2"><i><font color="Green" title="E8">A8</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2_4_1"><i><font color="Green" title="E15">A19</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_2_4_1_1_2"/>
( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_2_1_1_2_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:1_1_2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2_1_1_2"><i><font color="Green" title="E8">A8</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2_4_1"><i><font color="Green" title="E15">A19</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_4_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_2_4_1_1_3"/><i><font color="Green" title="E17">A21</font></i>: 
( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_2_1_1_2_4_1_1_3"/><b>then </b>
<font color="Maroon" title="c7">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z2</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2_4_1"><i><font color="Green" title="E14">A18</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2_4_1"><i><font color="Green" title="E16">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_2_1_1_2_4_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2_4_1"><i><font color="Green" title="E14">A18</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2_4_1"><i><font color="Green" title="E15">A19</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2_1_1_2_4_1_1_3"><i><font color="Green" title="E17">A21</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_4_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_2_4_1_1_4"/><i><font color="Green" title="E17">A22</font></i>: 
( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_2_1_1_2_4_1_1_4"/><b>then </b>
<font color="Maroon" title="c7">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z1</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2_1_1_2_4_1"><i><font color="Green" title="E13">A17</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2_4_1"><i><font color="Green" title="E16">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_2_1_1_2_4_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2_1_1_2_4_1"><i><font color="Green" title="E13">A17</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2_4_1"><i><font color="Green" title="E15">A19</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2_1_1_2_4_1_1_4"><i><font color="Green" title="E17">A22</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_4_1_1_5"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_2_4_1_1_5"/><i><font color="Green" title="E17">A23</font></i>: 
( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_2_1_1_2_4_1_1_5"/><b>then </b>
<font color="Maroon" title="c7">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z2</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2_4_1"><i><font color="Green" title="E14">A18</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2_4_1"><i><font color="Green" title="E16">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_2_1_1_2_4_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2_4_1"><i><font color="Green" title="E14">A18</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2_4_1"><i><font color="Green" title="E15">A19</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2_1_1_2_4_1_1_5"><i><font color="Green" title="E17">A23</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_4_1_1_6"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_2_4_1_1_6"/><i><font color="Green" title="E17">A24</font></i>: 
( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_2_1_1_2_4_1_1_6"/><b>then </b>
<font color="Maroon" title="c7">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z1</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2_1_1_2_4_1"><i><font color="Green" title="E13">A17</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2_4_1"><i><font color="Green" title="E16">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_2_1_1_2_4_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2_1_1_2_4_1"><i><font color="Green" title="E13">A17</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2_4_1"><i><font color="Green" title="E15">A19</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2_1_1_2_4_1_1_6"><i><font color="Green" title="E17">A24</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_1_1_2_4_1_1_7"><b>suppose </b></a><a NAME="E1:1_1_2_1_1_2_4_1_1_7"/><i><font color="Green" title="E17">A25</font></i>: 
( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c8">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_2_1_1_2_4_1_1_7"/><b>then </b><i><font color="Green" title="E18">A26</font></i>: 
<font color="Maroon" title="c7">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z1</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2_1_1_2_4_1"><i><font color="Green" title="E13">A17</font></i></a></i>;<br/><a NAME="E3:1_1_2_1_1_2_4_1_1_7"/>
<font color="Maroon" title="c7">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z2</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2_4_1"><i><font color="Green" title="E14">A18</font></i></a>, <a class="txt" href="sfmastr3.html#E1:1_1_2_1_1_2_4_1_1_7"><i><font color="Green" title="E17">A25</font></i></a></i>;<br/><b>hence </b><a NAME="E4:1_1_2_1_1_2_4_1_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:1_1_2_1_1_2_4_1"><i><font color="Green" title="E13">A17</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2_4_1"><i><font color="Green" title="E14">A18</font></i></a>, <a class="txt" href="sfmastr3.html#E3:1_1_2_1_1_2_4_1"><i><font color="Green" title="E15">A19</font></i></a>, <a class="txt" href="sfmastr3.html#E4:1_1_2_1_1_2_4_1"><i><font color="Green" title="E16">A20</font></i></a>, <a class="txt" href="sfmastr3.html#E2:1_1_2_1_1_2_4_1_1_7"><i><font color="Green" title="E18">A26</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:1_1_2_1_1_2_4"/>
<font color="Maroon" title="c7">p2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:1_1_2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a><br/>

<b>thus </b><a NAME="E3:1_1_2_1_1_2_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E5:1_1_2_1_1_2"><i><font color="Green" title="E12">A13</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
