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<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 <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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="stacks_1.html#D9" target="_self" title="STACKS_1:def.9">STACKS_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c2">S</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="c2">S</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><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">s</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="E2:80_2_1"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c2">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="c3">s</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="stacks_1.html#T34" target="_self" title="STACKS_1:th.34">Th70</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:80_2_1"/>
 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"> <font color="Maroon" title="c2">S</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="c2">S</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><br/>

<a NAME="E4:80_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A3</font></span>: 
 not  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c3">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_2_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="stacks_1.html#T36" target="_self" title="STACKS_1:th.36">Th71</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">P</font> =  <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="c3">s</font></span>)</span>) as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="stacks_1.html#T35" target="_self" title="STACKS_1:th.35">Th70a</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">E</font> =  <a href="stacks_1.html#K6" title="STACKS_1:func.6">top</a> <font color="Maroon" title="c3">s</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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> <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/>
<span class="kw">thus </span><font color="Maroon" title="c2">S</font> = 
 <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#K7" title="STACKS_1:func.7">push</a> (<span class="p2">(<span class="default"><a href="stacks_1.html#K6" title="STACKS_1:func.6">top</a> <font color="Maroon" title="c3">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="stacks_1.html#K5" title="STACKS_1:func.5">pop</a> <font color="Maroon" title="c3">s</font></span>)</span>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_2_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E4:80_2_1"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="stacks_1.html#D9" target="_self" title="STACKS_1:def.9">PUSHPOP</a></span>
<br/>.= 
 <a href="stacks_1.html#K7" title="STACKS_1:func.7">push</a> (<font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c4">P</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="stacks_1.html#T38" target="_self" title="STACKS_1:th.38">Th72</a></span>
<br/>.= 
 <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="c2">S</font></span>)</span>,<font color="Maroon" title="c4">P</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_2_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E4:80_2_1"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="stacks_1.html#T40" target="_self" title="STACKS_1:th.40">Th74</a></span>
<br/>.= 
 <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="c2">S</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="#E2:80_2_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E4:80_2_1"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="stacks_1.html#T39" target="_self" title="STACKS_1:th.39">Th73</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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