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

<b>set </b><font color="Maroon" title="c5">EE</font> =  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font>) <b>-&gt; </b>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font>, <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> =  <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,$1,<font color="Maroon" title="c4">LO</font>;<br/>
<a NAME="E1:15_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">s1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">s2</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">s1</font>) <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">s2</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">s1</font>, <font color="Maroon" title="c7">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font>;<br/>

<b>set </b><font color="Maroon" title="c8">f12</font> =  <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">s2</font></span>)</span>,<font color="Maroon" title="c4">LO</font>;<br/>
<b>set </b><font color="Maroon" title="c9">f1</font> =  <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c4">LO</font>;<br/>
<b>set </b><font color="Maroon" title="c10">f2</font> =  <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c4">LO</font>;<br/>
<a NAME="E1:15_1_1_1"/><i><font color="Green" title="E9">A3</font></i>: 
(  <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">s2</font></span>)</span>,<font color="Maroon" title="c4">LO</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font></span>)</span> &amp;  <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c4">LO</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font></span>)</span> &amp;  <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c4">LO</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/>
<a NAME="E2:15_1_1_1"/><i><font color="Green" title="E10">A4</font></i>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c4">LO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> )
 
<b>by </b><i><a class="txt" href="group_10.html#E1:15_1_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E3:15_1_1_1"/><i><font color="Green" title="E11">A5</font></i>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c4">LO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> )
 
<b>by </b><i><a class="txt" href="group_10.html#E1:15_1_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E4:15_1_1_1"/><i><font color="Green" title="E12">A6</font></i>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">s2</font></span>)</span>,<font color="Maroon" title="c4">LO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> )
 
<b>by </b><i><a class="txt" href="group_10.html#E1:15_1_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<div><i><font color="Green" title="E13">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:15_1_1_1_1_1"/><i><font color="Green" title="E14">A8</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">s2</font></span>)</span>,<font color="Maroon" title="c4">LO</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E2:15_1_1_1_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c4">LO</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E3:15_1_1_1"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/><a NAME="E3:15_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c4">LO</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</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> <span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c4">LO</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E3:15_1_1_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="group_10.html#E1:15_1_1_1_1_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c4">LO</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c4">LO</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E2:15_1_1_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:15_1_1_1_1"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c4">LO</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c4">LO</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c4">LO</font></span>)</span> )
 ;<br/><b>hence </b><a NAME="E3:15_1_1_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">s2</font></span>)</span>,<font color="Maroon" title="c4">LO</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E4:15_1_1_1"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_1_2"/><i><font color="Green" title="E14">A9</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">s2</font></span>)</span>,<font color="Maroon" title="c4">LO</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c12">X1</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> ;<br/><a NAME="E3:15_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c4">LO</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</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> <span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c4">LO</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E3:15_1_1_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="group_10.html#E1:15_1_1_1_2"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">X2</font> = <span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c4">LO</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> ;<br/><b>reconsider </b><font color="Maroon" title="c14">X3</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> <b>by </b><i><a class="txt" href="group_10.html#E1:15_1_1_1_2"><i><font color="Green" title="E14">A9</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">s2</font></span>)</span>,<font color="Maroon" title="c4">LO</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">s2</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c12">X1</font>
<b>by </b><i><a class="txt" href="group_10.html#E1:15"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="group_10.html#D4" target="_self" title="GROUP_10:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c7">s2</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c12">X1</font>
<b>by </b><i><a class="ref" href="group_10.html#D1" target="_self" title="GROUP_10:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c12">X1</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">X2</font>
<b>by </b><i><a class="txt" href="group_10.html#E1:15"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="group_10.html#D4" target="_self" title="GROUP_10:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c4">LO</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c4">LO</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E1:15"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="group_10.html#D4" target="_self" title="GROUP_10:def.4">Def4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:15_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">s1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">s2</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">s1</font>) <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">s2</font>)
 <b>by </b><i><a class="txt" href="group_10.html#E5:15_1_1_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15_1_1"/><i><font color="Green" title="E9">A10</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c3">S</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c6">f</font> =  <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c4">LO</font>;<br/>
<a NAME="E1:15_1_1_2"/>
 <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c4">LO</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/>
<a NAME="E2:15_1_1_2"/><b>then </b><i><font color="Green" title="E10">A11</font></i>: 
 ex <font color="Olive" title="b1">f'</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c4">LO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f'</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1"><b>now </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="E1:15_1_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">X</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> ;<br/><a NAME="E3:15_1_1_2_1"/><i><font color="Green" title="E11">A12</font></i>: 
<span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c4">LO</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="group_10.html#E1:15"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="group_10.html#D4" target="_self" title="GROUP_10:def.4">Def4</a></i>;<br/><a NAME="E4:15_1_1_2_1"/>
not  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="group_10.html#E1:15"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="group_10.html#T2" target="_self" title="GROUP_10:th.2">Th2</a></i>;<br/><a NAME="E5:15_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">X'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E7:15_1_1_2_1"/><i><font color="Green" title="E12">A13</font></i>: 
( <font color="Maroon" title="c9">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">X</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> )
 ;<br/><span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c4">LO</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">E</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">X'</font>
<b>by </b><i><a class="txt" href="group_10.html#E3:15_1_1_2_1"><i><font color="Green" title="E11">A12</font></i></a>, <a class="txt" href="group_10.html#E7:15_1_1_2_1"><i><font color="Green" title="E12">A13</font></i></a>, <a class="ref" href="group_10.html#D1" target="_self" title="GROUP_10:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c9">X'</font>
<b>by </b><i><a class="ref" href="funct_1.html#T162" title="FUNCT_1:th.162">FUNCT_1:162</a></i>
;<br/><b>hence </b><a NAME="E9:15_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c4">LO</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="group_10.html#E7:15_1_1_2_1"><i><font color="Green" title="E12">A13</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:15_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c3">S</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E2:15_1_1_2"><i><font color="Green" title="E10">A11</font></i></a>, <a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:15_1_1"/>
 ex <font color="Olive" title="b1">T</font> being  <a href="group_10.html#NM1" title="GROUP_10:NM.1">LeftOperation</a> of <font color="Maroon" title="c3">S</font>,<span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font></span>)</span> st <br/>for <font color="Olive" title="b2">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font> holds  <font color="Olive" title="b1">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">s</font>)
 
<b>from </b><i><a class="ref" href="group_10.html#S1" target="_self" title="GROUP_10:sch.1">GROUP_10:sch 1</a>(<a class="txt" href="group_10.html#E2:15_1_1"><i><font color="Green" title="E9">A10</font></i></a>, <a class="txt" href="group_10.html#E1:15_1_1"><i><font color="Green" title="E8">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:15_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="group_10.html#NM1" title="GROUP_10:NM.1">LeftOperation</a> of <font color="Maroon" title="c3">S</font>,<span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font></span>)</span> st <br/>for <font color="Olive" title="b2">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c2">n</font>,<font color="Olive" title="b2">s</font>,<font color="Maroon" title="c4">LO</font>
 ;<br/>


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