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<b>let </b><font color="Maroon" title="c2">IT1</font>, <font color="Maroon" title="c3">IT2</font> be  <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#V15" title="LEXBFS:attr.15">natural-v2labeled</a> <a href="lexbfs.html#NM9" title="LEXBFS:NM.9">VVGraphSeq</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:187_1_2"/><i><font color="Green" title="E101">A11</font></i>: 
( <font color="Maroon" title="c2">IT1</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#K20" title="LEXBFS:func.20">MCS:Init</a> <font color="Maroon" title="c1">G</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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#K25" title="LEXBFS:func.25">MCS:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:187_1_2"/><i><font color="Green" title="E102">A12</font></i>: 
( <font color="Maroon" title="c3">IT2</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#K20" title="LEXBFS:func.20">MCS:Init</a> <font color="Maroon" title="c1">G</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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#K25" title="LEXBFS:func.25">MCS:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> $1;<br/>
<a NAME="E3:187_1_2"/><i><font color="Green" title="E103">A13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="lexbfs.html#E1:187_1_2"><i><font color="Green" title="E101">A11</font></i></a>, <a class="txt" href="lexbfs.html#E2:187_1_2"><i><font color="Green" title="E102">A12</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="187_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:187_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ;<br/><b>then </b><font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> = 
 <a href="lexbfs.html#K25" title="LEXBFS:func.25">MCS:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="txt" href="lexbfs.html#E1:187_1_2"><i><font color="Green" title="E101">A11</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="lexbfs.html#E2:187_1_2"><i><font color="Green" title="E102">A12</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:187_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E5:187_1_2"/><b>then </b><i><font color="Green" title="E104">A14</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
;<br/>
<a NAME="E6:187_1_2"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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>]
 
<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#E3:187_1_2"><i><font color="Green" title="E103">A13</font></i></a>, <a class="txt" href="lexbfs.html#E5:187_1_2"><i><font color="Green" title="E104">A14</font></i></a>);<br/></i>
<a NAME="E7:187_1_2"/><b>then </b>
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c2">IT1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">IT2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
;<br/>
<b>hence </b><a NAME="E8:187_1_2"/>
<font color="Maroon" title="c2">IT1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">IT2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>


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