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<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">n</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c1">n</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> ;<br/>
<a NAME="E1:39_1_1"/><span class="lab"><font color="Green" title="E25">A1</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="ref" href="ordinal5.html#T13" target="_self" title="ORDINAL5:th.13">Th13</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E26">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">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>1</sub></font>[<font color="Maroon" title="c3">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:39_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">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>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">nk</font> = <font color="Maroon" title="c1">n</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c3">k</font> as   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/><a NAME="E3:39_1_1_1"/>
<font color="Maroon" title="c3">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>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">k</font>
 <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="E4:39_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</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">n</font>,<font color="Maroon" title="c4">nk</font>)
 <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="E5:39_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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>
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 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>]
 
<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="#E1:39_1_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:39_1_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:39_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c2">k</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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