<?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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c2">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies <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#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p1">(<span class="default"><span class="p2"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c2">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p1">(<span class="default"><span class="p2"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span><br/>

<a NAME="E3:13"/><i><font color="Green" title="E13">A3</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#K2" title="FUNCT_1:func.2">"</a>  <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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E2:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="functor0.html#T7" target="_self" title="FUNCTOR0:th.7">Th7</a></i>;<br/>
<a NAME="E4:13"/><b>then </b><i><font color="Green" title="E14">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></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="E5:13"/><i><font color="Green" title="E15">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">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/>
<a NAME="E6:13"/><i><font color="Green" title="E16">A6</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="E7:13"/><i><font color="Green" title="E17">A7</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> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E2:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="isocat_1.html#T1" title="ISOCAT_1:th.1">ISOCAT_1:1</a></i>;<br/>
<a NAME="E8:13"/><i><font color="Green" title="E18">A8</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E2:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="isocat_1.html#T1" title="ISOCAT_1:th.1">ISOCAT_1:1</a></i>;<br/>
<a NAME="E9:13"/><b>then </b><i><font color="Green" title="E19">A9</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>
 
<b>by </b><i><a class="ref" href="functor0.html#T9" target="_self" title="FUNCTOR0:th.9">Th9</a></i>;<br/>
<a NAME="E10:13"/><i><font color="Green" title="E20">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="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="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E2:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="functor0.html#T7" target="_self" title="FUNCTOR0:th.7">Th7</a></i>;<br/>
<i><font color="Green" title="E21">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_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2">(<span class="default"><span class="p3"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> = 
<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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="#E4:13"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="funct_4.html#T47" title="FUNCT_4:th.47">FUNCT_4:47</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</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 href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><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>
<b>by </b><i><a class="txt" href="#E8:13"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="#E10:13"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</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_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="#E5:13"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="funct_4.html#T48" title="FUNCT_4:th.48">FUNCT_4:48</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <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_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> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2">(<span class="default"><span class="p3"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> implies ( <font color="Maroon" title="c4">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> <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> &amp; <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> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> ) ) &amp; ( <font color="Maroon" title="c4">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> <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> &amp; <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> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> implies ( <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_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> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2">(<span class="default"><span class="p3"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> ) ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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> <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> &amp; <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> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> implies ( <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_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> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2">(<span class="default"><span class="p3"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> ) )</span>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:13_2_1"/><i><font color="Green" title="E22">A12</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#K2" title="RELAT_1:func.2">rng</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>and </b><br/><a NAME="E2:13_2_1"/><i><font color="Green" title="E23">A13</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2">(<span class="default"><span class="p3"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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> <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> &amp; <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> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> )</span><br/><a NAME="E3:13_2_1"/>
<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"><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>
 <b>by </b><i><a class="txt" href="#E5:13"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="#E1:13_2_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="funct_4.html#T48" title="FUNCT_4:th.48">FUNCT_4:48</a></i>;<br/><a NAME="E4:13_2_1"/><b>then </b>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</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#T88" title="FUNCT_3:th.88">FUNCT_3:88</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">y1</font>, <font color="Maroon" title="c6">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:13_2_1"/><i><font color="Green" title="E24">A14</font></i>: 
<font color="Maroon" title="c5">y1</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>and </b><br/><a NAME="E7:13_2_1"/><i><font color="Green" title="E25">A15</font></i>: 
<font color="Maroon" title="c6">y2</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="c2">g</font>
 <b>and </b><br/><a NAME="E8:13_2_1"/><i><font color="Green" title="E26">A16</font></i>: 
<font color="Maroon" title="c3">y</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">y1</font>,<font color="Maroon" title="c6">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/><b>set </b><font color="Maroon" title="c7">x1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y1</font>;<br/><b>set </b><font color="Maroon" title="c8">x2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y2</font>;<br/><a NAME="E9:13_2_1"/><i><font color="Green" title="E27">A17</font></i>: 
( <font color="Maroon" title="c6">y2</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="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> &amp; <font color="Maroon" title="c5">y1</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_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E2:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E6:13_2_1"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="#E7:13_2_1"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/><a NAME="E10:13_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c5">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E4:13"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then </b><i><font color="Green" title="E28">A18</font></i>: <span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2">(<span class="default"><span class="p3"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font> = 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></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">y2</font>,<font color="Maroon" title="c5">y1</font>
<b>by </b><i><a class="txt" href="#E3:13"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="funct_4.html#D2" title="FUNCT_4:def.2">FUNCT_4:def 2</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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E9:13_2_1"><i><font color="Green" title="E27">A17</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="E12:13_2_1"/>
( <font color="Maroon" title="c5">y1</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_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> &amp; <font color="Maroon" title="c6">y2</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="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E2:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E6:13_2_1"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="#E7:13_2_1"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/><a NAME="E13:13_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y1</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"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y2</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"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> )
 <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="E14:13_2_1"/><b>then </b><i><font color="Green" title="E29">A19</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y1</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; <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y2</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="c2">g</font> )
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E2:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><a NAME="E15:13_2_1"/><b>then </b><i><font color="Green" title="E30">A20</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><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="#E6:13"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E16:13_2_1"/><b>then </b><i><font color="Green" title="E31">A21</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><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="#E5:13"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="#E6:13"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="funct_4.html#T47" title="FUNCT_4:th.47">FUNCT_4:47</a></i>;<br/><b>thus </b><a NAME="E17:13_2_1"/>
<font color="Maroon" title="c4">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> <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="#E5:13"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="#E6:13"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="#E2:13_2_1"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="#E8:13_2_1"><i><font color="Green" title="E26">A16</font></i></a>, <a class="txt" href="#E11:13_2_1"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="#E15:13_2_1"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="funct_4.html#T47" title="FUNCT_4:th.47">FUNCT_4:47</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font></span><br/><a NAME="E18:13_2_1"/><i><font color="Green" title="E32">A22</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y1</font> &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y2</font> )
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E2:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E6:13_2_1"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="#E7:13_2_1"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/><b>thus </b><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> <font color="Maroon" title="c4">x</font> = 
<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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:13_2_1"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="#E8:13_2_1"><i><font color="Green" title="E26">A16</font></i></a>, <a class="txt" href="#E11:13_2_1"><i><font color="Green" title="E28">A18</font></i></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E16:13_2_1"><i><font color="Green" title="E31">A21</font></i></a>, <a class="ref" href="funct_4.html#T44" title="FUNCT_4:th.44">FUNCT_4:44</a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="#E8:13_2_1"><i><font color="Green" title="E26">A16</font></i></a>, <a class="txt" href="#E14:13_2_1"><i><font color="Green" title="E29">A19</font></i></a>, <a class="txt" href="#E18:13_2_1"><i><font color="Green" title="E32">A22</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><b>assume </b><b>that </b><br/><a NAME="E2:13_2"/><i><font color="Green" title="E22">A23</font></i>: 
<font color="Maroon" title="c4">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> <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>and </b><br/><a NAME="E3:13_2"/><i><font color="Green" title="E23">A24</font></i>: 
<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> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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_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> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2">(<span class="default"><span class="p3"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> )</span><br/><b>thus </b><a NAME="E4:13_2"/>
<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_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="#E2:13_2"><i><font color="Green" title="E22">A23</font></i></a>, <a class="txt" href="#E3:13_2"><i><font color="Green" title="E23">A24</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2">(<span class="default"><span class="p3"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font></span><br/><a NAME="E5:13_2"/>
<font color="Maroon" title="c4">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> <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="#E5:13"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="#E6:13"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="#E2:13_2"><i><font color="Green" title="E22">A23</font></i></a>, <a class="ref" href="funct_4.html#T47" title="FUNCT_4:th.47">FUNCT_4:47</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:13_2"/><i><font color="Green" title="E24">A25</font></i>: 
<font color="Maroon" title="c5">x1</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="c2">g</font>
 <b>and </b><br/><a NAME="E8:13_2"/><i><font color="Green" title="E25">A26</font></i>: 
<font color="Maroon" title="c6">x2</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="E9:13_2"/><i><font color="Green" title="E26">A27</font></i>: 
<font color="Maroon" title="c4">x</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">x1</font>,<font color="Maroon" title="c6">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E6:13"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/><i><font color="Green" title="E27">A28</font></i>: <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">x1</font>,<font color="Maroon" title="c6">x2</font> = 
<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">x2</font>,<font color="Maroon" title="c5">x1</font>
<b>by </b><i><a class="txt" href="#E2:13_2"><i><font color="Green" title="E22">A23</font></i></a>, <a class="txt" href="#E9:13_2"><i><font color="Green" title="E26">A27</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="c6">x2</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="c5">x1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E7:13_2"><i><font color="Green" title="E24">A25</font></i></a>, <a class="txt" href="#E8:13_2"><i><font color="Green" title="E25">A26</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="E11:13_2"/>
( <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</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="c2">g</font> &amp; <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</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="#E7:13_2"><i><font color="Green" title="E24">A25</font></i></a>, <a class="txt" href="#E8:13_2"><i><font color="Green" title="E25">A26</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:13_2"/><b>then </b>
<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">x1</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">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">g</font></span>)</span>,<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E13:13_2"/><b>then </b>
<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">x1</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">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><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="ref" href="funct_3.html#T88" title="FUNCT_3:th.88">FUNCT_3:88</a></i>;<br/><a NAME="E14:13_2"/><b>then </b><i><font color="Green" title="E28">A29</font></i>: 
<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">x1</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">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E7:13"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><a NAME="E15:13_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><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="#E6:13"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="#E7:13_2"><i><font color="Green" title="E24">A25</font></i></a>, <a class="txt" href="#E8:13_2"><i><font color="Green" title="E25">A26</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>hence </b><font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><span class="p2"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><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">x1</font>,<font color="Maroon" title="c6">x2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:13"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="#E9:13_2"><i><font color="Green" title="E26">A27</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<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">x1</font></span>)</span>,<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="c6">x2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:13_2"><i><font color="Green" title="E24">A25</font></i></a>, <a class="txt" href="#E8:13_2"><i><font color="Green" title="E25">A26</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">(<span class="default"><span class="p3"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <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="c6">x2</font></span>)</span>,<span class="p1">(<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">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E14:13_2"><i><font color="Green" title="E28">A29</font></i></a>, <a class="ref" href="funct_4.html#D2" title="FUNCT_4:def.2">FUNCT_4:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2">(<span class="default"><span class="p3"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <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="#E3:13_2"><i><font color="Green" title="E23">A24</font></i></a>, <a class="txt" href="#E9:13_2"><i><font color="Green" title="E26">A27</font></i></a>, <a class="txt" href="#E10:13_2"><i><font color="Green" title="E27">A28</font></i></a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:13"/>
<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#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p1">(<span class="default"><span class="p2"><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#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E9:13"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="#E11:13"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
