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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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">b</font> holds <br/><font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<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="c2">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</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"> (  <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="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">b</font> implies <font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<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="c2">a</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:83"/><span class="lab"><font color="Green" title="E59">A1</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="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <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="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<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="c2">a</font></span><br/>

<a NAME="E2:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A2</font></span>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/>
<a NAME="E3:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A3</font></span>: 
( 1 <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c2">a</font>)</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">b</font>,<font color="Maroon" title="c2">a</font>) &amp; 1 <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">n</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T39" title="ORDINAL2:th.39">ORDINAL2:39</a>, <a class="ref" href="ordinal2.html#T41" title="ORDINAL2:th.41">ORDINAL2:41</a></span>;<br/>
<a NAME="E4:83"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
(  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="ordinal4.html#T22" title="ORDINAL4:th.22">ORDINAL4:22</a></span>;<br/>
<a NAME="E5:83"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c2">a</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T31" title="ORDINAL3:th.31">ORDINAL3:31</a></span>;<br/>
<a NAME="E6:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A5</font></span>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">b</font> &amp;  <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="c3">n</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a>, <a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">c</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"> (  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c4">c</font>) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">n</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> implies  not <font color="Maroon" title="c4">c</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a> <font color="Maroon" title="c2">a</font> )</span><br/><span class="kw">assume </span><a NAME="E1:83_1"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
(  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c4">c</font>) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">n</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> &amp; <font color="Maroon" title="c4">c</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</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"> contradiction</span><br/><a NAME="E2:83_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</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="E3:83_1"/><span class="kw">then </span>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/><a NAME="E4:83_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A7</font></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> <font color="Maroon" title="c2">a</font></span>)</span>) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c4">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="ordinal4.html#T27" title="ORDINAL4:th.27">ORDINAL4:27</a></span>;<br/><a NAME="E5:83_1"/>
 <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> <font color="Maroon" title="c2">a</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c2">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a></span>;<br/><a NAME="E6:83_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">b</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">n</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c2">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E4:83_1"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E7:83_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="ordinal3.html#T35" title="ORDINAL3:th.35">ORDINAL3:35</a></span>;<br/><a NAME="E8:83_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E59">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:83_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>
<span class="kw">hence </span><a NAME="E8:83"/>
<font color="Maroon" title="c1">b</font> <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">b</font>,<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="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E6:83"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="ordinal5.html#D10" target="_self" title="ORDINAL5:def.10">Def10</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/>


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