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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">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#K6" title="STACKS_1:func.6">top</a> <font color="Olive" title="b2">S</font> <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="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">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#K6" title="STACKS_1:func.6">top</a> <font color="Olive" title="b1">S</font> <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="c2">s</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">t</font> =  the <a href="stacks_1.html#U4" title="STACKS_1:sel.4">s_top</a> of <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">A</font> =  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">set </span><font color="Maroon" title="c5">e</font> =  the    <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>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">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="c6">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#K6" title="STACKS_1:func.6">top</a> <font color="Maroon" title="c6">S</font> <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="c2">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:79"/><span class="lab"><font color="Green" title="E55">Z0</font></span>: 
<font color="Maroon" title="c6">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#K6" title="STACKS_1:func.6">top</a> <font color="Maroon" title="c6">S</font> <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="c2">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:79"/><span class="lab"><font color="Green" title="E56">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#K6" title="STACKS_1:func.6">top</a> <font color="Maroon" title="c6">S</font> <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="c2">s</font></span><br/>

<a NAME="E3:79"/><span class="kw">then </span>
 not  <a href="stacks_1.html#R1" title="STACKS_1:pred.1">emp</a> <font color="Maroon" title="c6">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E55">Z0</font></span></a>, <a class="ref" href="stacks_1.html#T36" target="_self" title="STACKS_1:th.36">Th71</a></span>;<br/>
<a NAME="E4:79"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<font color="Maroon" title="c6">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="stacks_1.html#T37" target="_self" title="STACKS_1:th.37">Th71a</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">f</font> =  the    <a href="orders_1.html#M1" title="ORDERS_1:mode.1">Choice_Function</a> of  <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>;<br/>
<a NAME="E5:79"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
( <font color="Maroon" title="c6">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> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</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="#E1:79"><span class="lab"><font color="Green" title="E55">Z0</font></span></a>, <a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></span>;<br/>
<a NAME="E6:79"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A3</font></span>: 
 the    <a href="orders_1.html#M1" title="ORDERS_1:mode.1">Choice_Function</a> of  <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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#D1" title="ORDERS_1:def.1">ORDERS_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> =  the    <a href="orders_1.html#M1" title="ORDERS_1:mode.1">Choice_Function</a> of  <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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">S</font> as   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E55">Z0</font></span></a></span>;<br/>
<a NAME="E8:79"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c8">x</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E55">Z0</font></span></a>, <a class="txt" href="#E6:79"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="ref" href="eqrel_1.html#T18" title="EQREL_1:th.18">EQREL_1:18</a></span>;<br/>
<a NAME="E9:79"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="stacks_1.html#R2" title="STACKS_1:pred.2">==</a> <font color="Maroon" title="c8">x</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/>
<a NAME="E10:79"/><span class="lab"><font color="Green" title="E61">A5</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the    <a href="orders_1.html#M1" title="ORDERS_1:mode.1">Choice_Function</a> of  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E5:79"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="ref" href="rlvect_3.html#T28" title="RLVECT_3:th.28">RLVECT_3:28</a></span>;<br/>
<a NAME="E11:79"/>
 the <a href="stacks_1.html#U4" title="STACKS_1:sel.4">s_top</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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="stacks_1.html#U4" title="STACKS_1:sel.4">s_top</a> of <font color="Maroon" title="c1">X</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a>  the    <a href="orders_1.html#M1" title="ORDERS_1:mode.1">Choice_Function</a> of  <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>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> ( the <a href="stacks_1.html#U1" title="STACKS_1:sel.1">s_empty</a> of <font color="Maroon" title="c1">X</font>, the    <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>)
 
<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">hence </span> <a href="stacks_1.html#K6" title="STACKS_1:func.6">top</a> <font color="Maroon" title="c6">S</font> = 
<span class="p1">(<span class="default"> the <a href="stacks_1.html#U4" title="STACKS_1:sel.4">s_top</a> of <font color="Maroon" title="c1">X</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a>  the    <a href="orders_1.html#M1" title="ORDERS_1:mode.1">Choice_Function</a> of  <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>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>
<br/>.= 
 <a href="stacks_1.html#K6" title="STACKS_1:func.6">top</a> <font color="Maroon" title="c8">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E10:79"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
 <a href="stacks_1.html#K6" title="STACKS_1:func.6">top</a> <font color="Maroon" title="c2">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:79"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E2:79"><span class="lab"><font color="Green" title="E56">Z1</font></span></a>, <a class="ref" href="stacks_1.html#T18" target="_self" title="STACKS_1:th.18">Th46</a></span>
;<br/> <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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