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<span class="kw">let </span><font color="Maroon" title="c1">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="Red">::  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> holds <br/><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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="Red">::  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="c2">n</font> implies <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:83"/><span class="lab"><font color="Green" title="E58">Z0</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">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font></span><br/>

<a NAME="E2:83"/><span class="kw">then </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="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></span>;<br/>
<a NAME="E3:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">Z1</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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">a</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a>, <a class="ref" href="ordinal2.html#T58" title="ORDINAL2:th.58">ORDINAL2:58</a></span>;<br/>
<a NAME="E4:83"/><span class="lab"><font color="Green" title="E60">Z5</font></span>: 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">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> 
 
<span class="kw">by</span> <span class="lab"><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><span class="lab"><font color="Green" title="E61">Z2</font></span>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &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="c2">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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">a</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a>, <a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E58">Z0</font></span></a>, <a class="ref" href="ordinal3.html#T34" title="ORDINAL3:th.34">ORDINAL3:34</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="c3">b</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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c3">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">a</font></span>)</span> implies  not <font color="Maroon" title="c3">b</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a> <font color="Maroon" title="c1">a</font> )</span><br/><span class="kw">assume </span><a NAME="E1:83_1"/><span class="lab"><font color="Green" title="E62">Z3</font></span>: 
(  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c3">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">a</font></span>)</span> &amp; <font color="Maroon" title="c3">b</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a> <font color="Maroon" title="c1">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:83_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</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="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></span>;<br/><a NAME="E4:83_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">Z4</font></span>: 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T61" title="ORDINAL2:th.61">ORDINAL2:61</a></span>;<br/><a NAME="E6:83_1"/><span class="kw">then </span>
<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">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="E62">Z3</font></span></a>, <a class="txt" href="#E4:83_1"><span class="lab"><font color="Green" title="E63">Z4</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>
(  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E60">Z5</font></span></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a>, <a class="ref" href="ordinal3.html#T38" title="ORDINAL3:th.38">ORDINAL3:38</a></span>;<br/><span class="kw">hence </span><a NAME="E8:83_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:83"/>
<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal5.html#K5" title="ORDINAL5:func.5">-exponent</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E59">Z1</font></span></a>, <a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E61">Z2</font></span></a>, <a class="ref" href="ordinal5.html#D10" target="_self" title="ORDINAL5:def.10">EXP</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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