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<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</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" title="c3">a</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">a</font></span><br/>

<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="c3">a</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">a</font>;<br/>
<font color="Maroon" title="c3">a</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  = 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a></i>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T47" title="ORDINAL2:th.47">ORDINAL2:47</a></i>
;<br/>
<a NAME="E2:80_1_1"/><b>then </b><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> ]
 
;<br/>
<div><i><font color="Green" title="E47">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">b</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="c5">b</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="c5">b</font>] )</span><br/><b>assume </b><a NAME="E1:80_1_1_2"/><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">b</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="c5">b</font>]</span><br/><b>defpred </b><font color="Maroon">S<sub>2</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><span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1</span>)</span>;<br/><span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a></i>
<br/>.= 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a></i>
;<br/><a NAME="E3:80_1_1_2"/><b>then </b><i><font color="Green" title="E49">A4</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 ;<br/><div><i><font color="Green" title="E50">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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>2</sub></font>[<font color="Maroon" title="c6">a</font>] implies <font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">a</font>] )</span><br/><b>assume </b><a NAME="E1:80_1_1_2_2"/><i><font color="Green" title="E51">A6</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">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>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">a</font>]</span><br/><span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">a</font></span>)</span> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c6">a</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T45" title="ORDINAL2:th.45">ORDINAL2:45</a></i>
<br/>.= 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ordinal3.html#E1:80_1_1_2_2"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="ordinal2.html#T45" title="ORDINAL2:th.45">ORDINAL2:45</a></i>
;<br/><b>hence </b><a NAME="E3:80_1_1_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">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="E5:80_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">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#E3:80_1_1_2"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="ordinal3.html#E4:80_1_1_2"><i><font color="Green" title="E50">A5</font></i></a>);<br/></i><b>hence </b><a NAME="E6:80_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">b</font>]
 <b>by </b><i><a class="txt" href="ordinal3.html#E1:80_1_1_2"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="ordinal2.html#T45" title="ORDINAL2:th.45">ORDINAL2:45</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="E4:80_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">b</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#E2:80_1_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="ordinal3.html#E3:80_1_1"><i><font color="Green" title="E47">A2</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/>


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