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<div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:21"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 0</span>)</span> = 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="ordinal2.html#T48">ORDINAL2:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> 0
<b>by </b><i><a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:21"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/>

<a NAME="E2:21_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="card_1.html#T52">CARD_1:52</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="card_4.html#E1:21_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="ordinal2.html#T45">ORDINAL2:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> 1
<b>by </b><i><a class="ref" href="ordinal2.html#T48">ORDINAL2:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K10">+^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T33">ORDINAL3:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="card_4.html#E2:21_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="ordinal2.html#T48">ORDINAL2:48</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:21"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="card_4.html#E1:21"><i><font color="Green">E17</font></i></a>, <a class="txt" href="card_4.html#E2:21"><i><font color="Green">E18</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:21"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>hence </b><a NAME="E5:21"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="card_1.html#T52">CARD_1:52</a></i>;<br/>


</div>
