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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<a NAME="E1:131"/><span class="lab"><font color="Green" title="E103">A1</font></span>: 
 <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#D10" onmouseover="rs('surrealo/D10')" onmouseout="rh()">SURREALO:def 10</a></span>;<br/>
<a NAME="E2:131"/>
(  <a href="surrealo.html#K2" title="SURREALO:func.2">born_eq</a> <span class="p1">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealo.html#K2" title="SURREALO:func.2">born_eq</a> <span class="p1">(<span class="default"><a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span> &amp;  <a href="surrealo.html#K2" title="SURREALO:func.2">born_eq</a> <span class="p1">(<span class="default"><a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font></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="ref" href="surrealo.html#T33" onmouseover="rs('surrealo/T33')" onmouseout="rh()">SURREALO:33</a>, <a class="ref" href="surrealo.html#D5" onmouseover="rs('surrealo/D5')" onmouseout="rh()">SURREALO:def 5</a>, <a class="ref" href="surrealo.html#D10" onmouseover="rs('surrealo/D10')" onmouseout="rh()">SURREALO:def 10</a>, <a class="ref" href="surrealn.html#T70" target="_self" onmouseover="rs('surrealn/T70')" onmouseout="rh()">Th70</a></span>;<br/>
<a NAME="E3:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A2</font></span>: 
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</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="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T48" onmouseover="rs('surrealo/T48')" onmouseout="rh()">SURREALO:48</a></span>;<br/>
<a NAME="E4:131"/>
<font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="131_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c2">B</font> =  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:131_1"/>
 not <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:131_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A3</font></span>: 
 <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p2">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T68" target="_self" onmouseover="rs('surrealn/T68')" onmouseout="rh()">Th68</a>, <a class="ref" href="ordinal1.html#T16" onmouseover="rs('ordinal1/T16')" onmouseout="rh()">ORDINAL1:16</a></span>;<br/>
<a NAME="E3:131_1"/>
 <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p2">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#D18" onmouseover="rs('surreal0/D18')" onmouseout="rh()">SURREAL0:def 18</a></span>;<br/>
<a NAME="E4:131_1"/><span class="kw">then </span>
 <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p2">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T69" target="_self" onmouseover="rs('surrealn/T69')" onmouseout="rh()">Th69</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:131_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131"><span class="lab"><font color="Green" title="E103">A1</font></span></a>, <a class="txt" href="#E2:131_1"><span class="lab"><font color="Green" title="E105">A3</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E5:131"/>
(  <a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealn.html#K8" title="SURREALN:func.8">No_Ordinal_op</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K9" title="SURREALN:func.9">No_uOrdinal_op</a> <font color="Maroon" title="c1">A</font></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="ref" href="surrealo.html#D10" onmouseover="rs('surrealo/D10')" onmouseout="rh()">SURREALO:def 10</a>, <a class="txt" href="#E3:131"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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