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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K39" title="GLIB_000:func.39">.Lifespan()</a>  holds <br/>( <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> <a href="tarski.html#R2" title="TARSKI: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> <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> &amp;  <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> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K39" title="GLIB_000:func.39">.Lifespan()</a>  holds <br/>( <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> <a href="tarski.html#R2" title="TARSKI: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">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> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">GS</font> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="Olive" title="b1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K39" title="GLIB_000:func.39">.Lifespan()</a>  implies ( <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="tarski.html#R2" title="TARSKI: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">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> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<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></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<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#K39" title="GLIB_000:func.39">.Lifespan()</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="tarski.html#R2" title="TARSKI: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">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> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<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></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">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#K17" title="FUNCOP_1:func.17">.--&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#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">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">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/>
<span class="kw">set </span><font color="Maroon" title="c6">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/>
<span class="kw">set </span><font color="Maroon" title="c7">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/>
<span class="kw">set </span><font color="Maroon" title="c8">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/>
<a NAME="E2:38"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K17" title="FUNCOP_1:func.17">.--&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#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">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>
 
;<br/>
<a NAME="E3:38"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<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="tarski.html#R2" title="TARSKI:pred.2">in</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E4:38"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<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#K17" title="FUNCOP_1:func.17">.--&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#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">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="lexbfs.html#D10" target="_self" onmouseover="rs('lexbfs/D10')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E5:38"/><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">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<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></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:38"/>
<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="tarski.html#R2" title="TARSKI: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">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<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></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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></span><br/>

<span class="kw">thus </span><a NAME="E7:38"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<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></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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