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<span class="kw">let </span><font color="Maroon" title="c1">b</font>, <font color="Maroon" title="c2">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"> ( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">b</font> implies <font color="Maroon" title="c2">a</font> <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">b</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:82"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">b</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="c2">a</font> <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">b</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>)</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:82_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><span class="kw">suppose </span></a><a NAME="E1:82_1_1"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</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="c2">a</font> <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">b</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>)</span><br/><div class="add"><span class="kw">assume </span><a NAME="E2:82_1_1"/>
 not <font color="Maroon" title="c2">a</font> <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">b</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:82_1_1"/><span class="kw">then </span>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T16" title="ORDINAL1:th.16">ORDINAL1:16</a></span>;<br/><a NAME="E4:82_1_1"/><span class="kw">then </span>
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E1:82_1_1"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="ordinal5.html#D10" target="_self" title="ORDINAL5:def.10">Def10</a>, <a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a></span>;<br/><a NAME="E5:82_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font>
 ;<br/><span class="kw">hence </span><a NAME="E6:82_1_1"/>
contradiction
 ; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2"><span class="kw">suppose </span></a><a NAME="E1:82_1_2"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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" title="c2">a</font> <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">b</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>)</span><br/><div class="add"><a NAME="E2:82_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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#D10" target="_self" title="ORDINAL5:def.10">Def10</a></span>;<br/><a NAME="E3:82_1_2"/><span class="kw">then </span>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T46" title="ORDINAL2:th.46">ORDINAL2:46</a></span>;<br/><span class="kw">hence </span><a NAME="E4:82_1_2"/>
<font color="Maroon" title="c2">a</font> <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">b</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E1:82_1_2"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="txt" href="ordinal5.html#E68"><span class="lab"><font color="Green" title="E56">Lm2</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></div><span class="kw">end;</span></div>

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