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<div class="add">

<b>reconsider </b><font color="Maroon" title="c1">N</font> = <font color="Maroon">F<sub>1</sub></font>() as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<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> ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">P<sub>1</sub></font>[$1,<font color="Olive" title="b1">n</font>];<br/>
<a NAME="E2:18_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">N</font>]
 
<b>by </b><i><a class="txt" href="nat_1.html#E1:18"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/>
<a NAME="E3:18_1"/><b>then </b><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E4:18_1"/><i><font color="Green" title="E7">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>given </b><font color="Maroon" title="c3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E2:18_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<b>reconsider </b><font color="Maroon" title="c4">F</font> = <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">n</font>) as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c4">F</font>]</span><br/>

<b>thus </b><a NAME="E4:18_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c4">F</font>]
 <b>by </b><i><a class="txt" href="nat_1.html#E1:18"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="nat_1.html#E2:18_1_1"><i><font color="Green" title="E8">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:18_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" target="_self" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="nat_1.html#E3:18_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="nat_1.html#E4:18_1"><i><font color="Green" title="E7">A3</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">n</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b3">m</font>] holds <br/><font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">m</font></span><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>for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">P<sub>1</sub></font>[$1,<font color="Olive" title="b1">n</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[$1,<font color="Olive" title="b2">m</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font>;<br/>
<a NAME="E6:18_1"/><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">n</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">m</font>] implies <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> )</span><br/>

<b>assume </b><a NAME="E1:18_1_2"/><i><font color="Green" title="E9">A6</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">n</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">m</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font></span><br/>

<a NAME="E2:18_1_2"/>
( ( for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">l</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">l</font>] or  not <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive" title="b2">l</font>) ) ) &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">l</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">l</font>] or  not <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive" title="b2">l</font>) ) ) )
 
;<br/>
<a NAME="E3:18_1_2"/><b>then </b>
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="txt" href="nat_1.html#E1:18"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="nat_1.html#E1:18_1_2"><i><font color="Green" title="E9">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:18_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:18_1"/><i><font color="Green" title="E9">A7</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:18_1_3"/><i><font color="Green" title="E10">A8</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>,<font color="Olive" title="b1">n</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>,<font color="Olive" title="b2">m</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<b>let </b><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c3">n</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c4">m</font>] implies <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> )</span><br/>

<b>assume </b><a NAME="E2:18_1_3"/><i><font color="Green" title="E11">A9</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c3">n</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c4">m</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">l</font>, <font color="Maroon" title="c6">u</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:18_1_3"/><i><font color="Green" title="E12">A10</font></i>: 
( <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c6">u</font>] &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">u</font>) )
 <b>by </b><i><a class="txt" href="nat_1.html#E1:18"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="nat_1.html#E2:18_1_3"><i><font color="Green" title="E11">A9</font></i></a></i>;<br/>
<a NAME="E5:18_1_3"/>
 ex <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>] &amp; <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b2">w</font>) )
 
<b>by </b><i><a class="txt" href="nat_1.html#E1:18"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="nat_1.html#E2:18_1_3"><i><font color="Green" title="E11">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:18_1_3"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:18_1_3"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="nat_1.html#E4:18_1_3"><i><font color="Green" title="E12">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E8:18_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" target="_self" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="nat_1.html#E6:18_1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="nat_1.html#E7:18_1"><i><font color="Green" title="E9">A7</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
