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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="stacks_1.html#NM3" title="STACKS_1:NM.3">StackAlgebra</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">s</font> being   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">x</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   ex <font color="Olive" title="b3">s1</font> being   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <font color="Maroon" title="c1">X</font> st <br/>( <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Olive" title="b3">s1</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> &amp; <font color="Olive" title="b3">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K11" title="STACKS_1:func.11">coset</a> <font color="Olive" title="b1">s</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">s</font> be   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <font color="Maroon" title="c1">X</font>; <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">x</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   ex <font color="Olive" title="b2">s1</font> being   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <font color="Maroon" title="c1">X</font> st <br/>( <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Olive" title="b2">s1</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K11" title="STACKS_1:func.11">coset</a> <font color="Maroon" title="c2">s</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">A</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>] <span class="kw">means </span> ex <font color="Olive" title="b1">s1</font> being   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <font color="Maroon" title="c1">X</font> st <br/>( <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Olive" title="b1">s1</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K11" title="STACKS_1:func.11">coset</a> <font color="Maroon" title="c2">s</font> );<br/>
<a NAME="E1:67"/>
 <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a>  <a href="stacks_1.html#K13" title="STACKS_1:func.13">core</a> <font color="Maroon" title="c2">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stacks_1.html#D19" target="_self" title="STACKS_1:def.19">CORE</a></span>;<br/>
<a NAME="E2:67"/><span class="kw">then </span>
<span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="stacks_1.html#K13" title="STACKS_1:func.13">core</a> <font color="Maroon" title="c2">s</font></span>)</span></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stacks_1.html#T5" target="_self" title="STACKS_1:th.5">Th31</a></span>;<br/>
<a NAME="E3:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stacks_1.html#T29" target="_self" title="STACKS_1:th.29">Th59a</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E45">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>; <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">x</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="c1">X</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="stacks_1.html#K1" title="STACKS_1:func.1">^</a> <font color="Maroon" title="c4">p</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <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="c1">X</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">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="stacks_1.html#K1" title="STACKS_1:func.1">^</a> <font color="Maroon" title="c4">p</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:67_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</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">S<sub>1</sub></font>[<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="stacks_1.html#K1" title="STACKS_1:func.1">^</a> <font color="Maroon" title="c4">p</font>]</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">s1</font> being   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E3:67_1"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
( <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">s1</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K11" title="STACKS_1:func.11">coset</a> <font color="Maroon" title="c2">s</font> )
 ;<br/><span class="kw">thus </span><a NAME="E4:67_1"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="stacks_1.html#K1" title="STACKS_1:func.1">^</a> <font color="Maroon" title="c4">p</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c7">s2</font> =  <a href="stacks_1.html#K7" title="STACKS_1:func.7">push</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">s1</font>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">s2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="stacks_1.html#K1" title="STACKS_1:func.1">^</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c7">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K11" title="STACKS_1:func.11">coset</a> <font color="Maroon" title="c2">s</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:67_1_1"/>
( <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">s2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="stacks_1.html#K1" title="STACKS_1:func.1">^</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c7">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K11" title="STACKS_1:func.11">coset</a> <font color="Maroon" title="c2">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="stacks_1.html#T8" target="_self" title="STACKS_1:th.8">Th34</a>, <a class="ref" href="stacks_1.html#D17" target="_self" title="STACKS_1:def.17">COSET</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></div><span class="kw">end;</span></div>
<a NAME="E5:67"/>
 for <font color="Olive" title="b1">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="stacks_1.html#S1" target="_self" title="STACKS_1:sch.1">STACKS_1:sch 1</a>(<a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E4:67"><span class="lab"><font color="Green" title="E45">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:67"/>
 for <font color="Olive" title="b1">x</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   ex <font color="Olive" title="b2">s1</font> being   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <font color="Maroon" title="c1">X</font> st <br/>( <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Olive" title="b2">s1</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K11" title="STACKS_1:func.11">coset</a> <font color="Maroon" title="c2">s</font> )
 ; <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>
