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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">S</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b2">n</font></span><br/><b>let </b><font color="Maroon" title="c2">GS</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">n</font></span><br/><b>let </b><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">n</font> )</span><br/>





<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">n</font></span><br/>

<b>set </b><font color="Maroon" title="c4">CSN</font> = <font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">n</font>;<br/>
<b>set </b><font color="Maroon" title="c5">VLN</font> = <font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">n</font>;<br/>
<b>set </b><font color="Maroon" title="c6">CN1</font> = <font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">VN1</font> = <font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">w</font> = <font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Maroon" title="c3">n</font>;<br/>
<a NAME="E2:42"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:42"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="lexbfs.html#D10" target="_self" title="LEXBFS:def.10">Def27</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">n</font></span>)</span>;<br/>
<a NAME="E3:42"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Maroon" title="c3">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E4:42"/><b>then </b><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:42"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>;<br/>
<b>hence </b><a NAME="E6:42"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K6" title="LEXBFS:func.6">.PickedAt</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="#E2:42"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="#E4:42"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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