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<b>let </b><font color="Maroon">IT1</font> be  <a href="glib_000.html#V14">finite</a> <a href="lexbfs.html#V11" target="_self">natural-vlabeled</a> <a href="lexbfs.html#V16" target="_self">finite-v2labeled</a> <a href="lexbfs.html#V17" target="_self">natsubset-v2labeled</a> <a href="lexbfs.html#NM9">VVGraphSeq</a>, <font color="Maroon">IT2</font> be  <a href="glib_000.html#V14">finite</a> <a href="lexbfs.html#V11" target="_self">natural-vlabeled</a> <a href="lexbfs.html#V16" target="_self">finite-v2labeled</a> <a href="lexbfs.html#V17" target="_self">natsubset-v2labeled</a> <a href="lexbfs.html#NM9">VVGraphSeq</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:141_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">IT1</font> <a href="lexbfs.html#K9" target="_self">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="lexbfs.html#K12" target="_self">LexBFS:Init</a> <font color="Maroon">G</font> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">IT1</font> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="lexbfs.html#K17" target="_self">LexBFS:Step</a> <span class="p1">(<span class="default"><font color="Maroon">IT1</font> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Olive">n</font></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:141_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">IT2</font> <a href="lexbfs.html#K9" target="_self">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="lexbfs.html#K12" target="_self">LexBFS:Init</a> <font color="Maroon">G</font> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">IT2</font> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="lexbfs.html#K17" target="_self">LexBFS:Step</a> <span class="p1">(<span class="default"><font color="Maroon">IT2</font> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Olive">n</font></span>)</span> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><font color="Maroon">IT1</font> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E3:141_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="lexbfs.html#D2" target="_self">Def2</a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:141_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>then </b><font color="Maroon">IT1</font> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
 <a href="lexbfs.html#K17" target="_self">LexBFS:Step</a> <span class="p1">(<span class="default"><font color="Maroon">IT2</font> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="lexbfs.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">IT2</font> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E3:141_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E5:141_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E6:141_1_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E7:141_1_2"/><b>then </b>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 
;<br/>
<b>hence </b><a NAME="E8:141_1_2"/>
<font color="Maroon">IT1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">IT2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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