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<span class="kw">let </span><font color="Maroon" title="c2">IT1</font>, <font color="Maroon" title="c3">IT2</font> be    <a href="lexbfs.html#M3" title="LEXBFS:mode.3">MCS:LabelingSeq</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"> ( <font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K18" title="LEXBFS:func.18">MCS:Init</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</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#K22" title="LEXBFS:func.22">MCS:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; <font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K18" title="LEXBFS:func.18">MCS:Init</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</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#K22" title="LEXBFS:func.22">MCS:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</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="E1:111_1_2"/><span class="lab"><font color="Green" title="E78">A7</font></span>: 
<font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K18" title="LEXBFS:func.18">MCS:Init</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E2:111_1_2"/><span class="lab"><font color="Green" title="E79">A8</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</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#K22" title="LEXBFS:func.22">MCS:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:111_1_2"/><span class="lab"><font color="Green" title="E80">A9</font></span>: 
<font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K18" title="LEXBFS:func.18">MCS:Init</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E4:111_1_2"/><span class="lab"><font color="Green" title="E81">A10</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</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#K22" title="LEXBFS:func.22">MCS:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</a> <font color="Olive" title="b1">n</font></span>)</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"> <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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><font color="Maroon" title="c2">IT1</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</a> $1;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_1_2_1"><span class="kw">now </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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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">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="E1:111_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</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"> <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#K23" title="LEXBFS:func.23">.</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#K22" title="LEXBFS:func.22">MCS:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:111_1_2"><span class="lab"><font color="Green" title="E79">A8</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">IT2</font> <a href="lexbfs.html#K23" title="LEXBFS:func.23">.</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="#E4:111_1_2"><span class="lab"><font color="Green" title="E81">A10</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E3:111_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" 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/></div><span class="kw">end;</span></div>
<a NAME="E6:111_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A11</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="E7:111_1_2"/><span class="lab"><font color="Green" title="E83">A12</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_2"><span class="lab"><font color="Green" title="E78">A7</font></span></a>, <a class="txt" href="#E3:111_1_2"><span class="lab"><font color="Green" title="E80">A9</font></span></a></span>;<br/>
<a NAME="E8:111_1_2"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E7:111_1_2"><span class="lab"><font color="Green" title="E83">A12</font></span></a>, <a class="txt" href="#E6:111_1_2"><span class="lab"><font color="Green" title="E82">A11</font></span></a>);<br/></span>
<a NAME="E9:111_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">object</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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="E10:111_1_2"/>
<font color="Maroon" title="c2">IT1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c3">IT2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" onmouseover="rs('pboole/T3')" onmouseout="rh()">PBOOLE: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"> verum</span><br/>


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