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<div class="add">

<div><span class="lab"><font color="Green" title="E25">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> : for <font color="Olive" title="b3">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o</font> holds <br/> ex <font color="Olive" title="b4">g</font>, <font color="Olive" title="b5">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b3">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">g</font> ) </span>}</span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">H</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">N</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">N</font> #) as   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="group_9.html#E20"><span class="lab"><font color="Green" title="E19">Lm7</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E2:38_1_1_1"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
not <font color="Maroon" title="c1">O</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">IT</font> being  <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of <font color="Maroon" title="c1">O</font>,<span class="p1">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span> st <br/>for <font color="Olive" title="b2">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font> holds  <font color="Olive" title="b1">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> :  ex <font color="Olive" title="b5">g</font>, <font color="Olive" title="b6">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">B</font> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b2">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">g</font> ) </span>}</span> </span><br/><a NAME="E3:38_1_1_1"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
 <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></span>;<br/><div><span class="lab"><font color="Green" title="E28">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font> implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</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_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span> )</span><br/><span class="kw">set </span><font color="Maroon" title="c6">f</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>);<br/><div><span class="lab"><font color="Green" title="E29">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>) implies  ex <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:38_1_1_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">A</font>, <font color="Maroon" title="c9">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font><span class="kw"> such that </span><br/><a NAME="E3:38_1_1_1_1_1"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c9">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:38_1_1_1_1_1"/>
for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> holds <br/> ex <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b1">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">A</font> = <font color="Maroon" title="c8">A</font>, <font color="Maroon" title="c11">B</font> = <font color="Maroon" title="c9">B</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c12">A</font> = <font color="Maroon" title="c10">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Olive" title="b1">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><span class="kw">take </span><font color="Maroon" title="c13">B</font> = <font color="Maroon" title="c11">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><span class="kw">thus </span><a NAME="E6:38_1_1_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:38_1_1_1_1"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</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_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c7">o</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1_1_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">y1</font>, <font color="Maroon" title="c10">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c10">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>) implies <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y2</font> )</span><br/><span class="kw">assume </span><a NAME="E2:38_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c10">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>) implies <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y2</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">A1</font>, <font color="Maroon" title="c12">B1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font><span class="kw"> such that </span><br/><a NAME="E4:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">A1</font>,<font color="Maroon" title="c12">B1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> holds <br/> ex <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A1</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B1</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b1">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> )
 ;<br/><span class="kw">assume </span><a NAME="E6:38_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c10">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">A2</font>, <font color="Maroon" title="c14">B2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font><span class="kw"> such that </span><br/><a NAME="E8:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c10">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">A2</font>,<font color="Maroon" title="c14">B2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> holds <br/> ex <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A2</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B2</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b1">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> )
 ;<br/><a NAME="E10:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A12</font></span>: 
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E11:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E12:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">f</font> = <font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font>;<br/><a NAME="E13:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">g1</font>, <font color="Maroon" title="c17">h1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E15:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<font color="Maroon" title="c16">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A1</font>
 <span class="kw">and </span><br/><a NAME="E16:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E40">A17</font></span>: 
<font color="Maroon" title="c17">h1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B1</font>
 <span class="kw">and </span><br/><a NAME="E17:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A18</font></span>: 
<font color="Maroon" title="c17">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A9</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">g2</font>, <font color="Maroon" title="c19">h2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E19:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E42">A19</font></span>: 
<font color="Maroon" title="c18">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A2</font>
 <span class="kw">and </span><br/><a NAME="E20:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E43">A20</font></span>: 
<font color="Maroon" title="c19">h2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B2</font>
 <span class="kw">and </span><br/><a NAME="E21:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A21</font></span>: 
<font color="Maroon" title="c19">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A11</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">A1</font> = <font color="Maroon" title="c11">A1</font>, <font color="Maroon" title="c21">A2</font> = <font color="Maroon" title="c13">A2</font>, <font color="Maroon" title="c22">B1</font> = <font color="Maroon" title="c12">B1</font>, <font color="Maroon" title="c23">B2</font> = <font color="Maroon" title="c14">B2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">H</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></span>;<br/><a NAME="E23:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A22</font></span>: 
<font color="Maroon" title="c21">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g2</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="group_9.html#E24"><span class="lab"><font color="Green" title="E23">Lm9</font></span></a></span>;<br/><a NAME="E24:38_1_1_1_1_2"/>
<font color="Maroon" title="c20">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="group_9.html#E24"><span class="lab"><font color="Green" title="E23">Lm9</font></span></a></span>;<br/><a NAME="E25:38_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">g1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E12:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E23:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E45">A22</font></span></a>, <a class="ref" href="group_2.html#T137" title="GROUP_2:th.137">GROUP_2:137</a></span>;<br/><a NAME="E26:38_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">g1</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="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E27:38_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">g1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E28:38_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">g2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="group_9.html#E25"><span class="lab"><font color="Green" title="E24">Lm10</font></span></a></span>;<br/><a NAME="E29:38_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">g2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></span>;<br/><a NAME="E30:38_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">h1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E21:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="ref" href="group_6.html#T41" title="GROUP_6:th.41">GROUP_6:41</a></span>;<br/><a NAME="E31:38_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">h1</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="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E32:38_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A23</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">h1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E33:38_1_1_1_1_2"/><span class="lab"><font color="Green" title="E47">A24</font></span>: 
<font color="Maroon" title="c23">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">h2</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="txt" href="group_9.html#E24"><span class="lab"><font color="Green" title="E23">Lm9</font></span></a></span>;<br/><a NAME="E34:38_1_1_1_1_2"/>
<font color="Maroon" title="c22">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">h1</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="txt" href="group_9.html#E24"><span class="lab"><font color="Green" title="E23">Lm9</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E35:38_1_1_1_1_2"/>
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E11:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E32:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">A23</font></span></a>, <a class="txt" href="#E33:38_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">A24</font></span></a>, <a class="ref" href="group_2.html#T137" title="GROUP_2:th.137">GROUP_2:137</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">f</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>) as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="funct_1.html#D1" title="FUNCT_1:def.1">FUNCT_1:def 1</a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">y1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> implies  ex <font color="Olive" title="b1">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y1</font>,<font color="Olive" title="b1">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> ) &amp; (  ex <font color="Olive" title="b1">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y1</font>,<font color="Olive" title="b1">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> implies <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y1</font>,<font color="Olive" title="b1">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> implies <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> )</span>
<div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c9">o</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1_1_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E2:38_1_1_1_1_3_1"/><span class="lab"><font color="Green" title="E31">A25</font></span>: 
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y1</font>,<font color="Olive" title="b1">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">A</font> = <font color="Maroon" title="c8">y1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> ;<br/><a NAME="E4:38_1_1_1_1_3_1"/>
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E31">A25</font></span></a>, <a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E6:38_1_1_1_1_3_1"/><span class="lab"><font color="Green" title="E32">A26</font></span>: 
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">and </span><br/><a NAME="E7:38_1_1_1_1_3_1"/>
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c11">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T15" title="GROUP_6:th.15">GROUP_6:15</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">h</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c9">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">B</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c9">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">H</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="group_2.html#D15" title="GROUP_2:def.15">GROUP_2:def 15</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">y2</font> = <font color="Maroon" title="c13">B</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c15">y2</font> = <font color="Maroon" title="c14">y2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c15">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_3_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">o</font> implies  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b2">h</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">B</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c16">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:38_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E33">A27</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b2">h</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">B</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c16">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c17">g</font> = <font color="Maroon" title="c11">g</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Maroon" title="c17">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">B</font> &amp; <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c16">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">g</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c18">h</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c9">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Maroon" title="c18">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">B</font> &amp; <font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c16">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">g</font> )</span><br/><span class="kw">thus </span><a NAME="E2:38_1_1_1_1_3_1_1"/>
<font color="Maroon" title="c17">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E6:38_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="txt" href="group_9.html#E24"><span class="lab"><font color="Green" title="E23">Lm9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">B</font> &amp; <font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c16">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">g</font> )</span><br/><span class="kw">thus </span><a NAME="E3:38_1_1_1_1_3_1_1"/>
<font color="Maroon" title="c18">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="group_9.html#E24"><span class="lab"><font color="Green" title="E23">Lm9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c16">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">g</font></span><br/><span class="kw">thus </span><a NAME="E4:38_1_1_1_1_3_1_1"/>
<font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c16">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E33">A27</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:38_1_1_1_1_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c15">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">given </span><font color="Maroon" title="c9">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E3:38_1_1_1_1_3"/><span class="lab"><font color="Green" title="E31">A28</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c10">A</font>, <font color="Maroon" title="c11">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font><span class="kw"> such that </span><br/><a NAME="E5:38_1_1_1_1_3"/><span class="lab"><font color="Green" title="E32">A29</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">A</font>,<font color="Maroon" title="c11">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:38_1_1_1_1_3"/>
for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> holds <br/> ex <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b1">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_1_1_3"><span class="lab"><font color="Green" title="E31">A28</font></span></a></span>;<br/><a NAME="E7:38_1_1_1_1_3"/>
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:38_1_1_1_1_3"/>
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38_1_1_1_1_3"><span class="lab"><font color="Green" title="E32">A29</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:38_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A30</font></span>: 
 <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>  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> implies <font color="Maroon" title="c8">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> )</span><br/><span class="kw">assume </span><a NAME="E1:38_1_1_1_1_4"/>
<font color="Maroon" title="c8">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:38_1_1_1_1_4"/><span class="lab"><font color="Green" title="E32">A31</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y1</font>,<font color="Maroon" title="c8">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">A</font>, <font color="Maroon" title="c11">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font><span class="kw"> such that </span><br/><a NAME="E5:38_1_1_1_1_4"/><span class="lab"><font color="Green" title="E33">A32</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y1</font>,<font color="Maroon" title="c8">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">A</font>,<font color="Maroon" title="c11">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:38_1_1_1_1_4"/>
for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> holds <br/> ex <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b1">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_1_1_4"><span class="lab"><font color="Green" title="E32">A31</font></span></a></span>;<br/><a NAME="E7:38_1_1_1_1_4"/>
<font color="Maroon" title="c11">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:38_1_1_1_1_4"/>
<font color="Maroon" title="c8">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38_1_1_1_1_4"><span class="lab"><font color="Green" title="E33">A32</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E8:38_1_1_1_1"/><span class="kw">then </span>
 <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="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E9:38_1_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</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_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:38_1_1_1_1"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:38_1_1_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">O</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span> st <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E4:38_1_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>);<br/></span><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">IT</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">O</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:38_1_1_1"/><span class="lab"><font color="Green" title="E29">A33</font></span>: 
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> <font color="Maroon" title="c1">O</font> holds <br/><font color="Maroon" title="c5">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">IT</font> = <font color="Maroon" title="c5">IT</font> as   <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of <font color="Maroon" title="c1">O</font>,<span class="p1">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span> ;<br/><span class="kw">take </span><font color="Maroon" title="c7">IT</font> = <font color="Maroon" title="c6">IT</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font> holds  <font color="Maroon" title="c7">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> :  ex <font color="Olive" title="b4">g</font>, <font color="Olive" title="b5">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b1">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">g</font> ) </span>}</span> </span><br/><span class="kw">let </span><font color="Maroon" title="c8">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> :  ex <font color="Olive" title="b3">g</font>, <font color="Olive" title="b4">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">g</font> ) </span>}</span> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">o</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">set </span><font color="Maroon" title="c10">X</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> : for <font color="Olive" title="b3">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o</font> holds <br/> ex <font color="Olive" title="b4">g</font>, <font color="Olive" title="b5">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b3">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">g</font> ) </span>}</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c11">Y</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> :  ex <font color="Olive" title="b3">g</font>, <font color="Olive" title="b4">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">g</font> ) </span>}</span> ;<br/><div><span class="lab"><font color="Green" title="E30">A34</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> : for <font color="Olive" title="b3">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o</font> holds <br/> ex <font color="Olive" title="b4">g</font>, <font color="Olive" title="b5">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b3">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">g</font> ) </span>}</span>  implies <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> :  ex <font color="Olive" title="b3">g</font>, <font color="Olive" title="b4">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">g</font> ) </span>}</span>  ) &amp; ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> :  ex <font color="Olive" title="b3">g</font>, <font color="Olive" title="b4">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">g</font> ) </span>}</span>  implies <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> : for <font color="Olive" title="b3">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o</font> holds <br/> ex <font color="Olive" title="b4">g</font>, <font color="Olive" title="b5">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b3">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">g</font> ) </span>}</span>  ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> :  ex <font color="Olive" title="b3">g</font>, <font color="Olive" title="b4">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">g</font> ) </span>}</span>  implies <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> : for <font color="Olive" title="b3">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o</font> holds <br/> ex <font color="Olive" title="b4">g</font>, <font color="Olive" title="b5">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b3">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">g</font> ) </span>}</span>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:38_1_1_1_2_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> : for <font color="Olive" title="b3">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o</font> holds <br/> ex <font color="Olive" title="b4">g</font>, <font color="Olive" title="b5">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b3">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">g</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> :  ex <font color="Olive" title="b3">g</font>, <font color="Olive" title="b4">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">g</font> ) </span>}</span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">A</font>, <font color="Maroon" title="c14">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font><span class="kw"> such that </span><br/><a NAME="E3:38_1_1_1_2_1"/><span class="lab"><font color="Green" title="E31">A35</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c14">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:38_1_1_1_2_1"/><span class="lab"><font color="Green" title="E32">A36</font></span>: 
for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> holds <br/> ex <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b1">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> )
 ;<br/><a NAME="E5:38_1_1_1_2_1"/>
 ex <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_1_1_2_1"><span class="lab"><font color="Green" title="E32">A36</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:38_1_1_1_2_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> :  ex <font color="Olive" title="b3">g</font>, <font color="Olive" title="b4">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">g</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_1_2_1"><span class="lab"><font color="Green" title="E31">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:38_1_1_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> :  ex <font color="Olive" title="b3">g</font>, <font color="Olive" title="b4">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">g</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> : for <font color="Olive" title="b3">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o</font> holds <br/> ex <font color="Olive" title="b4">g</font>, <font color="Olive" title="b5">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b3">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">g</font> ) </span>}</span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">A</font>, <font color="Maroon" title="c14">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font><span class="kw"> such that </span><br/><a NAME="E4:38_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A37</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c14">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:38_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A38</font></span>: 
 ex <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> )
 ;<br/><a NAME="E6:38_1_1_1_2"/>
for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> holds <br/> ex <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b1">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38_1_1_1_2"><span class="lab"><font color="Green" title="E32">A38</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:38_1_1_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> : for <font color="Olive" title="b3">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o</font> holds <br/> ex <font color="Olive" title="b4">g</font>, <font color="Olive" title="b5">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b3">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">g</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_1_1_2"><span class="lab"><font color="Green" title="E31">A37</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:38_1_1_1"/>
<font color="Maroon" title="c7">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> : for <font color="Olive" title="b3">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>  st <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o</font> holds <br/> ex <font color="Olive" title="b4">g</font>, <font color="Olive" title="b5">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b3">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">g</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E7:38_1_1_1"><span class="lab"><font color="Green" title="E29">A33</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E12:38_1_1_1"/>
<font color="Maroon" title="c7">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> :  ex <font color="Olive" title="b3">g</font>, <font color="Olive" title="b4">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">g</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:38_1_1_1"><span class="lab"><font color="Green" title="E30">A34</font></span></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:38_1_1_2"/>
<font color="Maroon" title="c1">O</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">IT</font> being  <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of <font color="Maroon" title="c1">O</font>,<span class="p1">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span> st <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p4">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">IT</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p4">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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 <font color="Maroon" title="c1">O</font>,<span class="p1">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="group_9.html#E7"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c5">IT</font> = <font color="Maroon" title="c4">IT</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p4">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><span class="kw">thus </span><a NAME="E3:38_1_1_2"/>
<font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p4">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:38_1_1"/>
( ( not <font color="Maroon" title="c1">O</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being  <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of <font color="Maroon" title="c1">O</font>,<span class="p1">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span> st <br/>for <font color="Olive" title="b2">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font> :  ex <font color="Olive" title="b5">g</font>, <font color="Olive" title="b6">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">B</font> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive" title="b2">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">g</font> ) </span>}</span>  ) &amp; ( <font color="Maroon" title="c1">O</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being  <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of <font color="Maroon" title="c1">O</font>,<span class="p1">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p4">(<span class="default"><a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
