<?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">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">S</font> being   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K14" title="STACKS_1:func.14">/==</a></span>)</span>  st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Olive" title="b1">s</font>) &amp;  not  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Olive" title="b1">s</font> holds <br/>(  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Olive" title="b2">S</font> &amp;  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Olive" title="b1">s</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Olive" title="b2">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">S</font> being   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K14" title="STACKS_1:func.14">/==</a></span>)</span>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c2">s</font>) &amp;  not  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c2">s</font> holds <br/>(  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Olive" title="b1">S</font> &amp;  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</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="c2">s</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Olive" title="b1">S</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">p</font> =  the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">i</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">S</font> be   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K14" title="STACKS_1:func.14">/==</a></span>)</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">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c2">s</font>) &amp;  not  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c2">s</font> implies (  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c5">S</font> &amp;  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</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="c2">s</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c5">S</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:78"/><span class="lab"><font color="Green" title="E54">Z0</font></span>: 
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font></span>)</span>,<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"> (  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c2">s</font> or (  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c5">S</font> &amp;  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</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="c2">s</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c5">S</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:78"/><span class="lab"><font color="Green" title="E55">Z1</font></span>: 
 not  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</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"> (  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c5">S</font> &amp;  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</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="c2">s</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c5">S</font> )</span><br/>

<a NAME="E3:78"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E54">Z0</font></span></a>, <a class="ref" href="eqrel_1.html#T20" title="EQREL_1:th.20">EQREL_1:20</a></span>;<br/>
<a NAME="E4:78"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E54">Z0</font></span></a>, <a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/>
<a NAME="E5:78"/><span class="lab"><font color="Green" title="E58">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></span>;<br/>
<a NAME="E6:78"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
 the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span> is    <a href="filter_1.html#M1" title="FILTER_1:mode.1">UnOp</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">X</font>, <a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">y1</font>, <font color="Maroon" title="c7">y2</font> be   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <font color="Maroon" title="c1">X</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_1.html#D1" title="FILTER_1:def.1">FILTER_1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:78_1"/><span class="lab"><font color="Green" title="E60">B1</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <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"> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:78_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A4</font></span>: 
<font color="Maroon" title="c6">y1</font> <a href="stacks_1.html#R2" title="STACKS_1:pred.2">==</a> <font color="Maroon" title="c7">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stacks_1.html#D16" target="_self" title="STACKS_1:def.16">REL</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:78_1_1"/>
(  not  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c6">y1</font> or  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c6">y1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:78_1_1_1"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
 not  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c6">y1</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="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><a NAME="E2:78_1_1_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">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="stacks_1.html#T14" target="_self" title="STACKS_1:th.14">Th42</a></span>;<br/><a NAME="E3:78_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">y1</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c7">y2</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1_1"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="ref" href="stacks_1.html#D3" target="_self" title="STACKS_1:def.3">EMP</a></span>;<br/><a NAME="E4:78_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A7</font></span>: 
( <span class="p1">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y1</font> &amp; <span class="p1">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:78"><span class="lab"><font color="Green" title="E58">A6</font></span></a>, <a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>;<br/><a NAME="E5:78_1_1_1"/>
 <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c6">y1</font> <a href="stacks_1.html#R2" title="STACKS_1:pred.2">==</a>  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c7">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E1:78_1_1_1"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="ref" href="stacks_1.html#T17" target="_self" title="STACKS_1:th.17">Th45</a></span>;<br/><span class="kw">hence </span><a NAME="E6:78_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_1_1_1"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="ref" href="stacks_1.html#D16" target="_self" title="STACKS_1:def.16">REL</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:78_1_1_2"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
 <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c6">y1</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="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><a NAME="E2:78_1_1_2"/><span class="kw">then </span>
 <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c7">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="stacks_1.html#T14" target="_self" title="STACKS_1:th.14">Th42</a></span>;<br/><a NAME="E3:78_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c6">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c7">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1_2"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="ref" href="stacks_1.html#D3" target="_self" title="STACKS_1:def.3">EMP</a></span>;<br/><a NAME="E4:78_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y1</font> &amp; <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font> &amp; <span class="p1">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y2</font> &amp; <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:78"><span class="lab"><font color="Green" title="E58">A6</font></span></a>, <a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>;<br/><span class="kw">hence </span><a NAME="E5:78_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1"><span class="lab"><font color="Green" title="E60">B1</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>

</div><span class="kw">end;</span></div>
<a NAME="E7:78"/><span class="lab"><font color="Green" title="E60">A8</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E55">Z1</font></span></a>, <a class="ref" href="stacks_1.html#D3" target="_self" title="STACKS_1:def.3">EMP</a></span>;<br/>
 <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c5">S</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="filter_1.html#K2" title="FILTER_1:func.2">/\/</a> <span class="p2">(<span class="default"><a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">S</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="stacks_1.html#D20" target="_self" title="STACKS_1:def.20">QUOT</a></span>
<br/>.= 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the <a href="stacks_1.html#U3" title="STACKS_1:sel.3">s_pop</a> of <font color="Maroon" title="c1">X</font> <a href="stacks_1.html#K4" title="STACKS_1:func.4">+*</a> <span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">s</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="ref" href="filter_1.html#D3" title="FILTER_1:def.3">FILTER_1:def 3</a></span>
<br/>.= 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</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="c2">s</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:78"><span class="lab"><font color="Green" title="E58">A6</font></span></a>, <a class="txt" href="#E7:78"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:78"/>
(  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c5">S</font> &amp;  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="stacks_1.html#K10" title="STACKS_1:func.10">==_</a> <font color="Maroon" title="c1">X</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="c2">s</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c5">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T20" title="EQREL_1:th.20">EQREL_1:20</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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