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

<b>reconsider </b><font color="Maroon" title="c3">IT</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as   <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">E</font> <b>by </b><i><a class="ref" href="group_9.html#T62" title="GROUP_9:th.62">GROUP_9:62</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">IT</font>
;<br/>

<div><i><font color="Green" title="E2">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_1_1_1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ;<br/><b>consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:4_1_1_1_1"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font>
 ;<br/><b>take </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">y</font>;<br/><a NAME="E4:4_1_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="group_10.html#E3:4_1_1_1_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:4_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">y</font></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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="group_10.html#E1:4_1_1_1_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E2:4_1_1_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T22" title="RELSET_1:th.22">RELSET_1:22</a></i>;<br/><a NAME="E3:4_1_1_1"/><b>then </b><i><font color="Green" title="E3">A4</font></i>: 
<font color="Maroon" title="c3">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">IT</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">IT'</font> = <font color="Maroon" title="c3">IT</font> as    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <b>by </b><i><a class="ref" href="relset_1.html#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/><a NAME="E5:4_1_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">IT'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><b>hence </b><a NAME="E6:4_1_1_1"/>
<font color="Maroon" title="c3">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font>
 <b>by </b><i><a class="txt" href="group_10.html#E3:4_1_1_1"><i><font color="Green" title="E3">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:4_1_1"/>
<font color="Maroon" title="c3">IT</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_10.html#D1" target="_self" title="GROUP_10:def.1">GROUP_10:def 1</a><br/>

<b>let </b><font color="Maroon" title="c4">s1</font>, <font color="Maroon" title="c5">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>

<a NAME="E4:4_1_1"/>
( <font color="Maroon" title="c3">IT</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Maroon" title="c3">IT</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c5">s2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font> )
 
<b>by </b><i><a class="txt" href="group_10.html#E2:4_1_1"><i><font color="Green" title="E2">A1</font></i></a></i>;<br/>
<a NAME="E5:4_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">IT</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">IT</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c5">s2</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T36" title="PARTFUN1:th.36">PARTFUN1:36</a></i>;<br/>
<b>hence </b><a NAME="E6:4_1_1"/>
<font color="Maroon" title="c3">IT</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">s2</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">IT</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">IT</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c5">s2</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E2:4_1_1"><i><font color="Green" title="E2">A1</font></i></a></i>;<br/>


</div>
