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

<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">s1</font>, <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">s2</font> being   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <font color="Maroon" title="c1">X</font>  st <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="Olive" title="b2">s</font> &amp; <font color="Olive" title="b3">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="Olive" title="b2">s</font> &amp; <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> <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Olive" title="b3">s2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> holds <br/><font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">s1</font>, <font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">s2</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"> ( <font color="Maroon" title="c2">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="c3">s</font> &amp; <font color="Maroon" title="c4">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="c3">s</font> &amp; <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">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> <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c4">s2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> implies <font color="Maroon" title="c2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> )</span><br/>







<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <font color="Maroon" title="c1">X</font>] <span class="kw">means </span> for <font color="Olive" title="b1">s2</font> being   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <font color="Maroon" title="c1">X</font>  st $1 <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="c3">s</font> &amp; <font color="Olive" title="b1">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="c3">s</font> &amp; <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default">$1</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="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Olive" title="b1">s2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> holds <br/>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s2</font>;<br/>
<a NAME="E1:69"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
 for <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  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Olive" title="b1">s1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">s1</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"> (  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c5">s1</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">s1</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_1"/><span class="lab"><font color="Green" title="E47">AA</font></span>: 
 <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c5">s1</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="c5">s1</font>]</span><br/>

<a NAME="E2:69_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c5">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>  <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/>
<span class="kw">let </span><font color="Maroon" title="c6">s2</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"> ( <font color="Maroon" title="c5">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="c3">s</font> &amp; <font color="Maroon" title="c6">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="c3">s</font> &amp; <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c5">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> <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">s2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> implies <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:69_1"/>
( <font color="Maroon" title="c5">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="c3">s</font> &amp; <font color="Maroon" title="c6">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="c3">s</font> &amp; <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c5">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> <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">s2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</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="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font></span><br/>

<a NAME="E4:69_1"/><span class="kw">then </span>
(  <a href="stacks_1.html#K13" title="STACKS_1:func.13">core</a> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stacks_1.html#K13" title="STACKS_1:func.13">core</a> <font color="Maroon" title="c3">s</font> &amp;  <a href="stacks_1.html#K13" title="STACKS_1:func.13">core</a> <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stacks_1.html#K13" title="STACKS_1:func.13">core</a> <font color="Maroon" title="c3">s</font> &amp;  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c6">s2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="stacks_1.html#T10" target="_self" title="STACKS_1:th.10">Th351</a>, <a class="ref" href="stacks_1.html#T31" target="_self" title="STACKS_1:th.31">Th14</a></span>;<br/>
<a NAME="E5:69_1"/><span class="kw">then </span>
(  <a href="stacks_1.html#K13" title="STACKS_1:func.13">core</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s1</font> &amp;  <a href="stacks_1.html#K13" title="STACKS_1:func.13">core</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E47">AA</font></span></a>, <a class="ref" href="stacks_1.html#T26" target="_self" title="STACKS_1:th.26">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:69_1"/>
<font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</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><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E47">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">s1</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">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">s1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="stacks_1.html#K7" title="STACKS_1:func.7">push</a> (<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c5">s1</font>)]</span><br/><span class="kw">let </span><font color="Maroon" title="c6">e</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c5">s1</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="stacks_1.html#K7" title="STACKS_1:func.7">push</a> (<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c5">s1</font>)] )</span><br/><span class="kw">assume </span><a NAME="E1:69_2"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">s1</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>[ <a href="stacks_1.html#K7" title="STACKS_1:func.7">push</a> (<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c5">s1</font>)]</span><br/><span class="kw">thus </span><a NAME="E2:69_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="stacks_1.html#K7" title="STACKS_1:func.7">push</a> (<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c5">s1</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="69_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">s2</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"> (  <a href="stacks_1.html#K7" title="STACKS_1:func.7">push</a> (<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c5">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="c3">s</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="c3">s</font> &amp; <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#K7" title="STACKS_1:func.7">push</a> (<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c5">s1</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> <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> implies  <a href="stacks_1.html#K7" title="STACKS_1:func.7">push</a> (<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c5">s1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_2_1"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
(  <a href="stacks_1.html#K7" title="STACKS_1:func.7">push</a> (<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c5">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="c3">s</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="c3">s</font> &amp; <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#K7" title="STACKS_1:func.7">push</a> (<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c5">s1</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> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="stacks_1.html#K7" title="STACKS_1:func.7">push</a> (<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c5">s1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font></span><br/>

<a NAME="E2:69_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A8</font></span>: 
<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="c6">e</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> <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c5">s1</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stacks_1.html#T8" target="_self" title="STACKS_1:th.8">Th34</a></span>;<br/>
<a NAME="E3:69_2_1"/><span class="kw">then </span>
 not  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c7">s2</font>
 
<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="E4:69_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A7</font></span>: 
<font color="Maroon" title="c7">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stacks_1.html#K7" title="STACKS_1:func.7">push</a> (<span class="p1">(<span class="default"><a href="stacks_1.html#K6" title="STACKS_1:func.6">top</a> <font color="Maroon" title="c7">s2</font></span>)</span>,<span class="p1">(<span class="default"><a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c7">s2</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stacks_1.html#D9" target="_self" title="STACKS_1:def.9">PUSHPOP</a></span>;<br/>
<a NAME="E5:69_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A9</font></span>: 
( <font color="Maroon" title="c5">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="c3">s</font> &amp;  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <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="c3">s</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_2_1"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="ref" href="stacks_1.html#T20" target="_self" title="STACKS_1:th.20">Th53</a></span>;<br/>
<a NAME="E6:69_2_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"><span class="p2">(<span class="default"><a href="stacks_1.html#K6" title="STACKS_1:func.6">top</a> <font color="Maroon" title="c7">s2</font></span>)</span></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> <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#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c7">s2</font></span>)</span></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_2_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="stacks_1.html#T8" target="_self" title="STACKS_1:th.8">Th34</a></span>;<br/>
<a NAME="E7:69_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stacks_1.html#K6" title="STACKS_1:func.6">top</a> <font color="Maroon" title="c7">s2</font> &amp; <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c5">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> <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#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c7">s2</font></span>)</span></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_2_1"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a>, <a class="ref" href="hilbert2.html#T2" title="HILBERT2:th.2">HILBERT2:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:69_2_1"/>
 <a href="stacks_1.html#K7" title="STACKS_1:func.7">push</a> (<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c5">s1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_2"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E4:69_2_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E5:69_2_1"><span class="lab"><font color="Green" title="E52">A9</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"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E3:69"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">s1</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="stacks_1.html#S3" target="_self" title="STACKS_1:sch.3">STACKS_1:sch 3</a>(<a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E47">A4</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:69"/>
( <font color="Maroon" title="c2">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="c3">s</font> &amp; <font color="Maroon" title="c4">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="c3">s</font> &amp; <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">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> <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c4">s2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> implies <font color="Maroon" title="c2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</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>
