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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> holds   <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b1">H</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K18" title="GROUP_2:func.18">index</a> <font color="Olive" title="b1">H</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">H</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c2">H</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">C</font> =  <a href="group_2.html#K15" title="GROUP_2:func.15">Left_Cosets</a> <font color="Maroon" title="c2">H</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> implies  ex <font color="Olive" title="b1">B</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">H</font> &amp; (  for <font color="Olive" title="b2">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">Y</font> holds <br/>( <font color="Maroon" title="c4">X</font>,<font color="Olive" title="b2">Y</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">Y</font> ) ) ) )</span><br/><span class="kw">assume </span><a NAME="E1:144_1"/><span class="lab"><font color="Green" title="E82">A1</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">H</font> &amp; (  for <font color="Olive" title="b2">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">Y</font> holds <br/>( <font color="Maroon" title="c4">X</font>,<font color="Olive" title="b2">Y</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">Y</font> ) ) )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E3:144_1"/>
<font color="Maroon" title="c5">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c4">X</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c6">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">H</font> &amp; (  for <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">Y</font> holds <br/>( <font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">Y</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">Y</font> ) ) )</span><br/><span class="kw">thus </span><a NAME="E5:144_1"/>
<font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">H</font> &amp; (  for <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">Y</font> holds <br/>( <font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">Y</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">Y</font> ) ) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E7:144_1"/><span class="lab"><font color="Green" title="E83">A2</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:144_1"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="group_2.html#D15" target="_self" title="GROUP_2:def.15">Def15</a></span>;<br/><a NAME="E8:144_1"/>
 ex <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c8">a</font> &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">B</font> &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T133" target="_self" title="GROUP_2:th.133">Th156</a></span>;<br/><span class="kw">hence </span><a NAME="E9:144_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:144_1"><span class="lab"><font color="Green" title="E83">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">Y</font> holds <br/>( <font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">Y</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">Y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">Y</font> implies ( <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c9">Y</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">Y</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E10:144_1"/><span class="lab"><font color="Green" title="E84">A3</font></span>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E11:144_1"/><span class="lab"><font color="Green" title="E85">A4</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c9">Y</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">Y</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">Y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:144_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a></span>;<br/><a NAME="E13:144_1"/><span class="lab"><font color="Green" title="E86">A5</font></span>: 
 ex <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:144_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="ref" href="group_2.html#D15" target="_self" title="GROUP_2:def.15">Def15</a></span>;<br/><span class="kw">hence </span><a NAME="E14:144_1"/>
<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c9">Y</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:144_1"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="group_2.html#T128" target="_self" title="GROUP_2:th.128">Th151</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">Y</font></span><br/><span class="kw">thus </span><a NAME="E15:144_1"/>
<font color="Maroon" title="c4">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:144_1"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E11:144_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E13:144_1"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="ref" href="group_2.html#T115" target="_self" title="GROUP_2:th.115">Th138</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:144"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A6</font></span>: 
 ex <font color="Olive" title="b1">D</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">C</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="group_2.html#E169"><span class="lab"><font color="Green" title="E80">Lm5</font></span></a></span>;<br/>
<a NAME="E4:144"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="group_2.html#K15" title="GROUP_2:func.15">Left_Cosets</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T137" target="_self" title="GROUP_2:th.137">Th167</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:144"/>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c2">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:144"><span class="lab"><font color="Green" title="E82">A6</font></span></a>, <a class="ref" href="group_2.html#D18" target="_self" title="GROUP_2:def.18">Def18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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