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<span class="kw">let </span><font color="Maroon" title="c1">a</font> be   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Olive" title="b2">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> implies <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c3">n</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:34"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c3">n</font></span><br/>

<a NAME="E2:34"/><span class="kw">then </span>
<font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:34"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></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="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/>
<a NAME="E5:34"/>
<font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T13" target="_self" title="ORDINAL5:th.13">Th13</a></span>;<br/>
<a NAME="E6:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A3</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="#E1:34"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="ordinal5.html#T16" target="_self" title="ORDINAL5:th.16">Th16</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E25">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" 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="c5">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:34_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>]
 ; <a class="txt" 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="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><a NAME="E2:34_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A5</font></span>: 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c5">n</font></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="ordinal4.html#T24" title="ORDINAL4:th.24">ORDINAL4:24</a></span>;<br/><a NAME="E3:34_1"/>
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T38" title="NAT_1:th.38">NAT_1:38</a></span>;<br/><a NAME="E4:34_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c5">n</font></span>)</span>) &amp; <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T14" target="_self" title="ORDINAL5:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E5:34_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a></span>; <a class="txt" 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="E8:34"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E7:34"><span class="lab"><font color="Green" title="E25">A4</font></span></a>);<br/></span>
<a NAME="E9:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<div><span class="lab"><font color="Green" title="E28">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">k</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:34_2"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><a NAME="E2:34_2"/>
<font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34"><span class="lab"><font color="Green" title="E26">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:34_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>; <a class="txt" 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="E11:34"/>
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E9:34"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E10:34"><span class="lab"><font color="Green" title="E28">A8</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E12:34"/>
<font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E23">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
