<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</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="lexbfs.html#NM2" title="LEXBFS:NM.2">VNumberingSeq</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b3">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Olive" title="b2">m</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">v</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">v</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="lexbfs.html#NM2" title="LEXBFS:NM.2">VNumberingSeq</a> of <font color="Maroon" title="c1">G</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">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Olive" title="b1">m</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">v</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">v</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font> ) )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">VLM</font> = <font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">j</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">VLJ</font> = <font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:52"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font> )</span><br/>

<a NAME="E2:52"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lexbfs.html#T20" target="_self" title="LEXBFS:th.20">Th20</a></span>;<br/>
<a NAME="E3:52"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="lexbfs.html#T15" target="_self" title="LEXBFS:th.15">Th15</a></span>;<br/>
<a NAME="E4:52"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lexbfs.html#T15" target="_self" title="LEXBFS:th.15">Th15</a></span>;<br/>
<a NAME="E5:52"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E6:52"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="xreal_1.html#T44" title="XREAL_1:th.44">XREAL_1:44</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E44">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_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="E1:52_1_1"/>
( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span> or <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font></span><br/><div class="add"><a NAME="E2:52_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lexbfs.html#T17" target="_self" title="LEXBFS:th.17">Th17</a></span>;<br/><a NAME="E3:52_1_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T11" title="RELAT_1:th.11">RELAT_1:11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:52_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E6:52"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="lexbfs.html#D10" target="_self" title="LEXBFS:def.10">Def10</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="52_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:52_1_1_2"/>
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:52_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</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="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_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="E1:52_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:52_2_1_1"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span><br/><div class="add"><a NAME="E2:52_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E7:52"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E3:52_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E4:52_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E5:52_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_2_1_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</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="52_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:52_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:52_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</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>
<span class="kw">hence </span><a NAME="E9:52"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.Lifespan()</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:52"><span class="lab"><font color="Green" title="E44">A6</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>
