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<span class="kw">set </span><font color="Maroon" title="c2">c</font> =  <a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E2:129_1_1"/><span class="lab"><font color="Green" title="E102">A1</font></span>: 
<font color="Maroon" title="c1">O</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="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="surrealn.html#E106"><span class="lab"><font color="Green" title="E100">Lm24</font></span></a></span>;<br/>
<a NAME="E3:129_1_1"/>
 <a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c1">O</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="E4:129_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A2</font></span>: 
 <a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</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="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:129_1_1"><span class="lab"><font color="Green" title="E102">A1</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<a NAME="E5:129_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A3</font></span>: 
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</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="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</font></span>)</span> &amp;  <a href="surrealo.html#K2" title="SURREALO:func.2">born_eq</a> <span class="p1">(<span class="default"><a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</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="c3">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="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#D5" onmouseover="rs('surrealo/D5')" onmouseout="rh()">SURREALO:def 5</a>, <a class="ref" href="surrealo.html#T48" onmouseover="rs('surrealo/T48')" onmouseout="rh()">SURREALO:48</a>, <a class="ref" href="surrealn.html#T70" target="_self" onmouseover="rs('surrealn/T70')" onmouseout="rh()">Th70</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:129_1_1"/>
 not  <a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</font> is  <a href="surrealn.html#V3" title="SURREALN:attr.3">No_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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E8:129_1_1"/><span class="lab"><font color="Green" title="E105">A4</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1">(<span class="default"><a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">C</font> = <font color="Maroon" title="c4">C</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:129_1_1"><span class="lab"><font color="Green" title="E105">A4</font></span></a>, <a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a></span>;<br/>
<a NAME="E10:129_1_1"/>
<font color="Maroon" title="c5">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2">(<span class="default"><a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2">(<span class="default"><a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:129_1_1"><span class="lab"><font color="Green" title="E105">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E11:129_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A5</font></span>: 
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c5">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a></span>;<br/>
<a NAME="E12:129_1_1"/>
<font color="Maroon" title="c5">C</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> <font color="Maroon" title="c5">C</font></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="E13:129_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">C</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> <font color="Maroon" title="c5">C</font></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/>
<a NAME="E14:129_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">C</font> <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="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:129_1_1"><span class="lab"><font color="Green" title="E106">A5</font></span></a>, <a class="ref" href="surrealn.html#T68" target="_self" onmouseover="rs('surrealn/T68')" onmouseout="rh()">Th68</a>, <a class="txt" href="#E5:129_1_1"><span class="lab"><font color="Green" title="E104">A3</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<a NAME="E15:129_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A6</font></span>: 
<font color="Maroon" title="c5">C</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a>  <a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:129_1_1"><span class="lab"><font color="Green" title="E103">A2</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<a NAME="E16:129_1_1"/>
(  <a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1">(<span class="default"><a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <font color="Maroon" title="c1">O</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="surrealo.html#T11" onmouseover="rs('surrealo/T11')" onmouseout="rh()">SURREALO:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:129_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:129_1_1"><span class="lab"><font color="Green" title="E105">A4</font></span></a>, <a class="txt" href="#E15:129_1_1"><span class="lab"><font color="Green" title="E107">A6</font></span></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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