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

<b>set </b><font color="Maroon" title="c6">X</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">k1</font>,<font color="Olive" title="b2">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> : <font color="Olive" title="b1">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a>  </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">k1</font>,<font color="Olive" title="b2">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k1</font> <a href="commacat.html#K5" title="COMMACAT:func.5">*</a> <font color="Olive" title="b2">k2</font> );<br/>
<a NAME="E1:8_1_1"/><i><font color="Green" title="E6">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">k1</font>,<font color="Olive" title="b3">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b2">k1</font>, <font color="Olive" title="b3">k2</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> : <font color="Olive" title="b2">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a>  </span>}</span>  holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:8_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">k1</font>,<font color="Olive" title="b2">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> : <font color="Olive" title="b1">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">k1</font>, <font color="Maroon" title="c9">k2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font><b> such that </b><br/><a NAME="E3:8_1_1_1"/><i><font color="Green" title="E7">A2</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">k1</font>,<font color="Maroon" title="c9">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c8">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a>  )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c8">k1</font> <a href="commacat.html#K5" title="COMMACAT:func.5">*</a> <font color="Maroon" title="c9">k2</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c10">y</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c8">k1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c9">k2</font>
;<br/>

<b>thus </b><a NAME="E5:8_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">k1</font>,<font color="Maroon" title="c9">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k1</font> <a href="commacat.html#K5" title="COMMACAT:func.5">*</a> <font color="Maroon" title="c9">k2</font> )
 <b>by </b><i><a class="txt" href="commacat.html#E3:8_1_1_1"><i><font color="Green" title="E7">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:8_1_1"/><i><font color="Green" title="E7">A3</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">k1</font>,<font color="Olive" title="b2">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> : <font color="Olive" title="b1">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a>  </span>}</span>  &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">k1</font>,<font color="Olive" title="b3">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b2">k1</font>, <font color="Olive" title="b3">k2</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> : <font color="Olive" title="b2">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a>  </span>}</span>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="commacat.html#E1:8_1_1"><i><font color="Green" title="E6">A1</font></i></a>);<br/></i>
<a NAME="E4:8_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:8_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">f</font>
 ;<br/>

<a NAME="E2:8_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> st <br/>( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">k1</font>,<font color="Olive" title="b2">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b1">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a>  )
 
<b>by </b><i><a class="txt" href="commacat.html#E3:8_1_1"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:8_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:8_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:8_1_1_3"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:8_1_1_3"/><i><font color="Green" title="E9">A5</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</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="E4:8_1_1_3"/>
 ex <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> st <br/>( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">k1</font>,<font color="Olive" title="b2">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k1</font> <a href="commacat.html#K5" title="COMMACAT:func.5">*</a> <font color="Olive" title="b2">k2</font> )
 
<b>by </b><i><a class="txt" href="commacat.html#E3:8_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="commacat.html#E3:8_1_1_3"><i><font color="Green" title="E9">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:8_1_1_3"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="commacat.html#E3:8_1_1_3"><i><font color="Green" title="E9">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">f</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>, <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> <b>by </b><i><a class="txt" href="commacat.html#E4:8_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">f</font>
;<br/>

<b>thus </b><a NAME="E7:8_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">k1</font>,<font color="Olive" title="b2">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> : <font color="Olive" title="b1">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a>  </span>}</span> 
 <b>by </b><i><a class="txt" href="commacat.html#E3:8_1_1"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c9">k1</font>, <font color="Maroon" title="c10">k2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>;<br/>



<b>assume </b><a NAME="E8:8_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">k1</font>,<font color="Maroon" title="c10">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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> <font color="Maroon" title="c8">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">k</font>, <font color="Maroon" title="c12">k'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font><b> such that </b><br/><a NAME="E10:8_1_1"/><i><font color="Green" title="E9">A6</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">k1</font>,<font color="Maroon" title="c10">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c12">k'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">k1</font>,<font color="Maroon" title="c10">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="commacat.html#K5" title="COMMACAT:func.5">*</a> <font color="Maroon" title="c12">k'</font> )
 <b>by </b><i><a class="txt" href="commacat.html#E3:8_1_1"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>
<a NAME="E11:8_1_1"/>
( <font color="Maroon" title="c9">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c10">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k'</font> )
 
<b>by </b><i><a class="txt" href="commacat.html#E10:8_1_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E12:8_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">k1</font>,<font color="Maroon" title="c10">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k1</font> <a href="commacat.html#K5" title="COMMACAT:func.5">*</a> <font color="Maroon" title="c10">k2</font>
 <b>by </b><i><a class="txt" href="commacat.html#E10:8_1_1"><i><font color="Green" title="E9">A6</font></i></a></i>;<br/>


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