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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b><font color="Maroon" title="c1">a</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">b</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>;<br/>
<a NAME="E1:78_2_1"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="ref" href="ordinal2.html#T52" title="ORDINAL2:th.52">ORDINAL2:52</a></i>;<br/>
<div><i><font color="Green" title="E47">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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="c3">a</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">a</font>] )</span><br/><b>assume </b><a NAME="E1:78_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">a</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>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">a</font>]</span><br/><b>then reconsider </b><font color="Maroon" title="c4">c</font> = <font color="Maroon" title="c3">a</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">b</font> as  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><a NAME="E3:78_2_1_1"/>
<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T53" title="ORDINAL2:th.53">ORDINAL2:53</a></i>;<br/><b>hence </b><a NAME="E4:78_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">a</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="E3:78_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">a</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S17" title="ORDINAL2:sch.17">ORDINAL2:sch 17</a>(<a class="txt" href="ordinal3.html#E1:78_2_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="ordinal3.html#E2:78_2_1"><i><font color="Green" title="E47">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:78_2_1"/>
<font color="Maroon" title="c1">a</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">b</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
