<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">IT1</font>, <font color="Maroon" title="c3">IT2</font> be    <a href="lexbfs.html#M2" title="LEXBFS:mode.2">LexBFS:LabelingSeq</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K9" title="LEXBFS:func.9">LexBFS: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#K13" title="LEXBFS:func.13">.</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K12" title="LEXBFS:func.12">LexBFS:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; <font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K9" title="LEXBFS:func.9">LexBFS: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#K13" title="LEXBFS:func.13">.</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K12" title="LEXBFS:func.12">LexBFS:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) implies <font color="Maroon" title="c2">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">IT2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:70_1_2"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
<font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K9" title="LEXBFS:func.9">LexBFS:Init</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E2:70_1_2"/><span class="lab"><font color="Green" title="E49">A7</font></span>: 
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#K13" title="LEXBFS:func.13">.</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K12" title="LEXBFS:func.12">LexBFS:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:70_1_2"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
<font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K9" title="LEXBFS:func.9">LexBFS:Init</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E4:70_1_2"/><span class="lab"><font color="Green" title="E51">A9</font></span>: 
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#K13" title="LEXBFS:func.13">.</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K12" title="LEXBFS:func.12">LexBFS:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">IT2</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> $1;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:70_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><span class="kw">then </span><font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</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="lexbfs.html#K12" title="LEXBFS:func.12">LexBFS:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_2"><span class="lab"><font color="Green" title="E49">A7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_2"><span class="lab"><font color="Green" title="E51">A9</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E3:70_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:70_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A10</font></span>: 
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#K2" title="NAT_1:func.2">+</a> 1]
 
;<br/>
<a NAME="E7:70_1_2"/><span class="lab"><font color="Green" title="E53">A11</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_2"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E3:70_1_2"><span class="lab"><font color="Green" title="E50">A8</font></span></a></span>;<br/>
<a NAME="E8:70_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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E7:70_1_2"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E6:70_1_2"><span class="lab"><font color="Green" title="E52">A10</font></span></a>);<br/></span>
<a NAME="E9:70_1_2"/><span class="kw">then </span>
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/>
<span class="kw">hence </span><a NAME="E10:70_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
