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<div class="add">

<b>set </b><font color="Maroon" title="c2">E</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</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="c1">S</font>) <b>-&gt; </b>   <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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> =  <a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> $1;<br/>
<a NAME="E1:9_1_1"/><i><font color="Green" title="E3">A1</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="c1">S</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">s1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">s2</font>) <a href="funct_2.html#R2" title="FUNCT_2: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="9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">s1</font>, <font color="Maroon" title="c4">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/>

<b>set </b><font color="Maroon" title="c5">f12</font> =  <a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">s2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">f1</font> =  <a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c3">s1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">f2</font> =  <a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c4">s2</font>;<br/>
<a NAME="E1:9_1_1_1"/><i><font color="Green" title="E4">A2</font></i>: 
(  <a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp;  <a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c3">s1</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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp;  <a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c4">s2</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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,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="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/>
<a NAME="E2:9_1_1_1"/><b>then </b><i><font color="Green" title="E5">A3</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#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</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> &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> 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="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E3:9_1_1_1"/><i><font color="Green" title="E6">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#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</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> &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> 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="txt" href="group_10.html#E1:9_1_1_1"><i><font color="Green" title="E4">A2</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="E4:9_1_1_1"/><i><font color="Green" title="E7">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#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</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> &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> 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="txt" href="group_10.html#E1:9_1_1_1"><i><font color="Green" title="E4">A2</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="E8">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1"><b>now </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> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_1"/><i><font color="Green" title="E9">A7</font></i>: 
<font color="Maroon" title="c8">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#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">s2</font></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E2:9_1_1_1_1_1"/>
<font color="Maroon" title="c8">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#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c4">s2</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E3:9_1_1_1"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/><a NAME="E3:9_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c4">s2</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#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#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c4">s2</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E3:9_1_1_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="group_10.html#E1:9_1_1_1_1_1"><i><font color="Green" title="E9">A7</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="E4:9_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c4">s2</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#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#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E2:9_1_1_1"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:9_1_1_1_1"/>
( <font color="Maroon" title="c8">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#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c4">s2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c4">s2</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#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#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> )
 ;<br/><b>hence </b><a NAME="E3:9_1_1_1_1"/>
<font color="Maroon" title="c8">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#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">s2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E4:9_1_1_1"><i><font color="Green" title="E7">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_2"><b>now </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> ;<br/><b>assume </b><a NAME="E1:9_1_1_1_2"/><i><font color="Green" title="E9">A8</font></i>: 
<font color="Maroon" title="c8">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#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">s2</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">s1'</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/><a NAME="E3:9_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c4">s2</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#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#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c4">s2</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E3:9_1_1_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="group_10.html#E1:9_1_1_1_2"><i><font color="Green" title="E9">A8</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="c10">s1''</font> = <span class="p1">(<span class="default"><a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">s2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">s1'</font>
<b>by </b><i><a class="ref" href="topgrp_1.html#D1" title="TOPGRP_1:def.1">TOPGRP_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">s1'</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">s1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">s1''</font>
<b>by </b><i><a class="ref" href="topgrp_1.html#D1" title="TOPGRP_1:def.1">TOPGRP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c3">s1</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#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<b>by </b><i><a class="ref" href="topgrp_1.html#D1" title="TOPGRP_1:def.1">TOPGRP_1:def 1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:9_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">s1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">s2</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">s1</font>) <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">s2</font>)
 <b>by </b><i><a class="txt" href="group_10.html#E5:9_1_1_1"><i><font color="Green" title="E8">A6</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="E2:9_1_1"/><i><font color="Green" title="E4">A9</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">S</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c3">f</font> =  <a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</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>;<br/>
<a NAME="E1:9_1_1_2"/>
 <a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,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="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/>
<a NAME="E2:9_1_1_2"/><b>then </b><i><font color="Green" title="E5">A10</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#NK3" title="GROUP_10:NK.3">the_left_translation_of</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f'</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f'</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> &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> 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="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="9_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_1_2_1"/>
<font color="Maroon" title="c4">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>then reconsider </b><font color="Maroon" title="c5">s1</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/><a NAME="E3:9_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <span class="p2">(<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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">s1</font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#D1" title="TOPGRP_1:def.1">TOPGRP_1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:9_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <span class="p2">(<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></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="c4">x</font>
 <b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:9_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">S</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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="txt" href="group_10.html#E2:9_1_1_2"><i><font color="Green" title="E5">A10</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="E3:9_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="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> 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="c1">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#E2:9_1_1"><i><font color="Green" title="E4">A9</font></i></a>, <a class="txt" href="group_10.html#E1:9_1_1"><i><font color="Green" title="E3">A1</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:9_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="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> 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="c1">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#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Olive" title="b2">s</font>
 ;<br/>


</div>
