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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="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">a</font> implies 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</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="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:135"/><span class="lab"><font color="Green" title="E74">Z0</font></span>: 
1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</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="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</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></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">b</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E3:135"/><span class="lab"><font color="Green" title="E75">A0</font></span>: 
1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</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="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T75" target="_self" title="ORDINAL6:th.75">Th13</a>, <a class="ref" href="ordinal5.html#T51" title="ORDINAL5:th.51">ORDINAL5:51</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">c</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E5:135"/><span class="lab"><font color="Green" title="E76">A1</font></span>: 
 <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</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> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T76" target="_self" title="ORDINAL6:th.76">Th14</a></span>;<br/>
<a NAME="E6:135"/><span class="lab"><font color="Green" title="E77">A2</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a></span>;<br/>
<a NAME="E7:135"/>
 <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a>, <a class="ref" href="ordinal5.html#T44" title="ORDINAL5:th.44">ORDINAL5:44</a></span>;<br/>
<a NAME="E8:135"/><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">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:135"><span class="lab"><font color="Green" title="E74">Z0</font></span></a>, <a class="txt" href="#E5:135"><span class="lab"><font color="Green" title="E76">A1</font></span></a>, <a class="ref" href="ordinal6.html#T72" target="_self" title="ORDINAL6:th.72">Th15a</a></span>;<br/>
<a NAME="E9:135"/><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">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/>
<span class="kw">hence </span><a NAME="E10:135"/>
1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</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="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:135"><span class="lab"><font color="Green" title="E76">A1</font></span></a>, <a class="ref" href="ordinal6.html#T71" target="_self" title="ORDINAL6:th.71">Th15</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</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></span><br/>

<span class="kw">assume </span><a NAME="E11:135"/>
 <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</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="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</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 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="E12:135"/><span class="kw">then </span>
 <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:135"><span class="lab"><font color="Green" title="E75">A0</font></span></a>, <a class="ref" href="ordinal6.html#T4" target="_self" title="ORDINAL6:th.4">EXCH1</a></span>;<br/>
<a NAME="E13:135"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T12" target="_self" title="ORDINAL6:th.12">LemE2</a></span>;<br/>
<a NAME="E14:135"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<a NAME="E15:135"/><span class="kw">then </span>
 <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T11" target="_self" title="ORDINAL6:th.11">LemE1</a></span>;<br/>
<a NAME="E16:135"/><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="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:135"><span class="lab"><font color="Green" title="E74">Z0</font></span></a>, <a class="txt" href="#E3:135"><span class="lab"><font color="Green" title="E75">A0</font></span></a>, <a class="ref" href="ordinal6.html#T71" target="_self" title="ORDINAL6:th.71">Th15</a></span>;<br/>
<a NAME="E17:135"/><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="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:135"><span class="lab"><font color="Green" title="E77">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:135"/>
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/>


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