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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( $2 is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a> &amp; $1 is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> &amp;  ex <font color="Olive" title="b1">nn</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ex <font color="Olive" title="b2">Gn</font>, <font color="Olive" title="b3">Gn1</font> being <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">nn</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Gn</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">Gn1</font> &amp; <font color="Olive" title="b3">Gn1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K17" title="LEXBFS:func.17">LexBFS:Step</a> <font color="Olive" title="b2">Gn</font> ) ) or ( (  not $2 is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a> or  not $1 is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ) &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 ) );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:141_1_1_1_1_1"/>
( ( <font color="Maroon" title="c3">x</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a> &amp; <font color="Maroon" title="c2">n</font> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ) or  not <font color="Maroon" title="c3">x</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a> or  not <font color="Maroon" title="c2">n</font> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:141_1_1_1_1_1_1"/><i><font color="Green" title="E66">A1</font></i>: 
( <font color="Maroon" title="c3">x</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a> &amp; <font color="Maroon" title="c2">n</font> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">Gn</font> = <font color="Maroon" title="c3">x</font> as  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a> ;<br/><b>reconsider </b><font color="Maroon" title="c5">nn</font> = <font color="Maroon" title="c2">n</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="lexbfs.html#E1:141_1_1_1_1_1_1"><i><font color="Green" title="E66">A1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">xx</font> = <font color="Maroon" title="c3">x</font> as  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a> <b>by </b><i><a class="txt" href="lexbfs.html#E1:141_1_1_1_1_1_1"><i><font color="Green" title="E66">A1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">SGn</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a><b> such that </b><br/><a NAME="E6:141_1_1_1_1_1_1"/><i><font color="Green" title="E67">A2</font></i>: 
<font color="Maroon" title="c7">SGn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K17" title="LEXBFS:func.17">LexBFS:Step</a> <font color="Maroon" title="c4">Gn</font>
 ;<br/><b>thus </b><a NAME="E7:141_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y</font>]
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:141_1_1_1_1_1_1"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="lexbfs.html#E6:141_1_1_1_1_1_1"><i><font color="Green" title="E67">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:141_1_1_1_1_1_2"/>
(  not <font color="Maroon" title="c3">x</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a> or  not <font color="Maroon" title="c2">n</font> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:141_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:141_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y</font>]
 ;<br/></div><b>end;</b></div>
<a NAME="E2:141_1_1"/><b>then </b><i><font color="Green" title="E66">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
;<br/>
<a NAME="E3:141_1_1"/><i><font color="Green" title="E67">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b4">y2</font>] holds <br/><font color="Olive" title="b3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y2</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c2">IT</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:141_1_1"/><i><font color="Green" title="E68">A5</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c2">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K12" title="LEXBFS:func.12">LexBFS:Init</a> <font color="Maroon" title="c1">G</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<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>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S1" title="RECDEF_1:sch.1">RECDEF_1:sch 1</a>(<a class="txt" href="lexbfs.html#E2:141_1_1"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="lexbfs.html#E3:141_1_1"><i><font color="Green" title="E67">A4</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c3">IT</font> = <font color="Maroon" title="c2">IT</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="lexbfs.html#E5:141_1_1"><i><font color="Green" title="E68">A5</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c3">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a>;<br/>
<a NAME="E7:141_1_1"/><i><font color="Green" title="E69">A6</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="txt" href="lexbfs.html#E5:141_1_1"><i><font color="Green" title="E68">A5</font></i></a></i>;<br/>
<div><i><font color="Green" title="E70">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:141_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">n</font>]
 ;<br/><b>then consider </b><font color="Maroon" title="c5">nn</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>, <font color="Maroon" title="c6">Gn</font>, <font color="Maroon" title="c7">Gn1</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a><b> such that </b><br/><a NAME="E3:141_1_1_2"/><i><font color="Green" title="E71">A8</font></i>: 
( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nn</font> &amp; <font color="Maroon" title="c3">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Gn</font> &amp; <font color="Maroon" title="c3">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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> <font color="Maroon" title="c7">Gn1</font> &amp; <font color="Maroon" title="c7">Gn1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K17" title="LEXBFS:func.17">LexBFS:Step</a> <font color="Maroon" title="c6">Gn</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E5:141_1_1"><i><font color="Green" title="E68">A5</font></i></a></i>;<br/><b>thus </b><a NAME="E4:141_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="lexbfs.html#E3:141_1_1_2"><i><font color="Green" title="E71">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:141_1_1"/><i><font color="Green" title="E71">A9</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="lexbfs.html#E7:141_1_1"><i><font color="Green" title="E69">A6</font></i></a>, <a class="txt" href="lexbfs.html#E8:141_1_1"><i><font color="Green" title="E70">A7</font></i></a>);<br/></i>
<a NAME="E10:141_1_1"/><b>then </b>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">IT</font> = <font color="Maroon" title="c3">IT</font> as   <a href="glib_000.html#NM11" title="GLIB_000:NM.11">GraphSeq</a> <b>by </b><i><a class="ref" href="glib_000.html#D55" title="GLIB_000:def.55">GLIB_000:def 55</a></i>;<br/>
<a NAME="E12:141_1_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c4">IT</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Olive" title="b1">n</font> is <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a> &amp; <font color="Maroon" title="c4">IT</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Olive" title="b1">n</font> is <a href="lexbfs.html#V4" title="LEXBFS:attr.4">[V2Labeled]</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E1:141_1_1_3"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E2:141_1_1_3"/>
( <font color="Maroon" title="c4">IT</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">n</font> is <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a> &amp; <font color="Maroon" title="c4">IT</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">n</font> is <a href="lexbfs.html#V4" title="LEXBFS:attr.4">[V2Labeled]</a> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E9:141_1_1"><i><font color="Green" title="E71">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">IT</font> = <font color="Maroon" title="c4">IT</font> as   <a href="lexbfs.html#NM9" title="LEXBFS:NM.9">VVGraphSeq</a> <b>by </b><i><a class="ref" href="glib_003.html#D26" title="GLIB_003:def.26">GLIB_003:def 26</a>, <a class="ref" href="lexbfs.html#D20" target="_self" title="LEXBFS:def.20">Def20</a></i>;<br/>
<a NAME="E14:141_1_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c5">IT</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Olive" title="b1">n</font> is <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> &amp; <font color="Maroon" title="c5">IT</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Olive" title="b1">n</font> is <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> &amp; <font color="Maroon" title="c5">IT</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Olive" title="b1">n</font> is <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> &amp; <font color="Maroon" title="c5">IT</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Olive" title="b1">n</font> is <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E1:141_1_1_4"/>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E2:141_1_1_4"/>
( <font color="Maroon" title="c5">IT</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c6">n</font> is <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> &amp; <font color="Maroon" title="c5">IT</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c6">n</font> is <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> &amp; <font color="Maroon" title="c5">IT</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c6">n</font> is <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> &amp; <font color="Maroon" title="c5">IT</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c6">n</font> is <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E9:141_1_1"><i><font color="Green" title="E71">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">IT</font> = <font color="Maroon" title="c5">IT</font> as  <a href="glib_000.html#V14" title="GLIB_000:attr.14">finite</a> <a href="lexbfs.html#V11" title="LEXBFS:attr.11">natural-vlabeled</a> <a href="lexbfs.html#V16" title="LEXBFS:attr.16">finite-v2labeled</a> <a href="lexbfs.html#V17" title="LEXBFS:attr.17">natsubset-v2labeled</a> <a href="lexbfs.html#NM9" title="LEXBFS:NM.9">VVGraphSeq</a> <b>by </b><i><a class="ref" href="glib_000.html#D60" title="GLIB_000:def.60">GLIB_000:def 60</a>, <a class="ref" href="lexbfs.html#D17" target="_self" title="LEXBFS:def.17">Def17</a>, <a class="ref" href="lexbfs.html#D23" target="_self" title="LEXBFS:def.23">Def23</a>, <a class="ref" href="lexbfs.html#D22" target="_self" title="LEXBFS:def.22">Def22</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">IT</font>
;<br/>

<b>thus </b><a NAME="E16:141_1_1"/>
<font color="Maroon" title="c6">IT</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K12" title="LEXBFS:func.12">LexBFS:Init</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E5:141_1_1"><i><font color="Green" title="E68">A5</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E17:141_1_1"/><i><font color="Green" title="E72">A10</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">nn</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>, <font color="Maroon" title="c9">Gn</font>, <font color="Maroon" title="c10">Gn1</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a><b> such that </b><br/><a NAME="E19:141_1_1"/><i><font color="Green" title="E73">A11</font></i>: 
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">nn</font> &amp; <font color="Maroon" title="c6">IT</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Gn</font> &amp; <font color="Maroon" title="c6">IT</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Gn1</font> &amp; <font color="Maroon" title="c10">Gn1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K17" title="LEXBFS:func.17">LexBFS:Step</a> <font color="Maroon" title="c9">Gn</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E5:141_1_1"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="lexbfs.html#E17:141_1_1"><i><font color="Green" title="E72">A10</font></i></a></i>;<br/>
<b>thus </b><a NAME="E20:141_1_1"/>
<font color="Maroon" title="c6">IT</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K17" title="LEXBFS:func.17">LexBFS:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">IT</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E19:141_1_1"><i><font color="Green" title="E73">A11</font></i></a></i>;<br/>


</div>
