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

<b>let </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>

<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E10">A1</font></i>: 
 <a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<a NAME="E2:12"/><i><font color="Green" title="E11">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E3:12"/><i><font color="Green" title="E12">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">g</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/>
<b>assume </b><a NAME="E4:12"/>
<font color="Maroon" title="c3">x1</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x11</font>, <font color="Maroon" title="c6">x12</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:12"/><i><font color="Green" title="E13">A4</font></i>: 
( <font color="Maroon" title="c5">x11</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> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c6">x12</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> <font color="Maroon" title="c1">f</font> )
 <b>and </b><br/><a NAME="E7:12"/><i><font color="Green" title="E14">A5</font></i>: 
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x11</font>,<font color="Maroon" title="c6">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="functor0.html#E2:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/>
<b>assume </b><a NAME="E8:12"/>
<font color="Maroon" title="c4">x2</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x21</font>, <font color="Maroon" title="c8">x22</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:12"/><i><font color="Green" title="E15">A6</font></i>: 
( <font color="Maroon" title="c7">x21</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> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c8">x22</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> <font color="Maroon" title="c1">f</font> )
 <b>and </b><br/><a NAME="E11:12"/><i><font color="Green" title="E16">A7</font></i>: 
<font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x21</font>,<font color="Maroon" title="c8">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="functor0.html#E2:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/>
<a NAME="E12:12"/>
<font color="Maroon" title="c3">x1</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 
<b>by </b><i><a class="txt" href="functor0.html#E2:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="functor0.html#E6:12"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="functor0.html#E7:12"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E13:12"/><b>then </b><i><font color="Green" title="E17">A8</font></i>: 
<font color="Maroon" title="c3">x1</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"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="functor0.html#E2:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="functor0.html#E3:12"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="funct_4.html#T47" title="FUNCT_4:th.47">FUNCT_4:47</a></i>;<br/>
<a NAME="E14:12"/>
<font color="Maroon" title="c4">x2</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 
<b>by </b><i><a class="txt" href="functor0.html#E2:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="functor0.html#E10:12"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="functor0.html#E11:12"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E15:12"/><b>then </b><i><font color="Green" title="E18">A9</font></i>: 
<font color="Maroon" title="c4">x2</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"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="functor0.html#E2:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="functor0.html#E3:12"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="funct_4.html#T47" title="FUNCT_4:th.47">FUNCT_4:47</a></i>;<br/>
<b>assume </b><a NAME="E16:12"/><i><font color="Green" title="E19">A10</font></i>: 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x2</font>
 ;<br/>

<a NAME="E17:12"/>
( <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">x11</font>,<font color="Maroon" title="c6">x12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x11</font></span>)</span>,<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="c6">x12</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">x21</font>,<font color="Maroon" title="c8">x22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x21</font></span>)</span>,<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="c8">x22</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="functor0.html#E6:12"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="functor0.html#E10:12"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E18:12"/><b>then </b><i><font color="Green" title="E20">A11</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="c8">x22</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="c6">x12</font> &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x21</font> )
 
<b>by </b><i><a class="txt" href="functor0.html#E16:12"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a>, <a class="txt" href="functor0.html#E7:12"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="functor0.html#E11:12"><i><font color="Green" title="E16">A7</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x11</font>,<font color="Maroon" title="c6">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">x11</font>,<font color="Maroon" title="c6">x12</font>

<br/>.= 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">x12</font>,<font color="Maroon" title="c5">x11</font>
<b>by </b><i><a class="txt" href="functor0.html#E7:12"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="functor0.html#E13:12"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="funct_4.html#T44" title="FUNCT_4:th.44">FUNCT_4:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</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="c8">x22</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x21</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="functor0.html#E6:12"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="functor0.html#E18:12"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">x22</font>,<font color="Maroon" title="c7">x21</font>
<b>by </b><i><a class="txt" href="functor0.html#E10:12"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">x21</font>,<font color="Maroon" title="c8">x22</font>
<b>by </b><i><a class="txt" href="functor0.html#E11:12"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="functor0.html#E15:12"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="funct_4.html#T44" title="FUNCT_4:th.44">FUNCT_4:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x21</font>,<font color="Maroon" title="c8">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

;<br/>
<b>hence </b><a NAME="E20:12"/>
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font>
 <b>by </b><i><a class="txt" href="functor0.html#E1:12"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="functor0.html#E13:12"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="functor0.html#E15:12"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a>, <a class="txt" href="functor0.html#E7:12"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="functor0.html#E11:12"><i><font color="Green" title="E16">A7</font></i></a></i>;<br/>


</div>
