<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  iff <font color="Maroon" title="c1">r</font> is  not   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a> )</span><br/>

<a NAME="E1:84"/>
<a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D7" target="_self" onmouseover="rs('surrealn/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E2:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A1</font></span>: 
 <a href="surrealo.html#K2" title="SURREALO:func.2">born_eq</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</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#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span>
 
<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="ref" href="surrealo.html#T33" onmouseover="rs('surrealo/T33')" onmouseout="rh()">SURREALO:33</a></span>;<br/>
<a NAME="E3:84"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
 <a href="surrealo.html#K2" title="SURREALO:func.2">born_eq</a> <span class="p1">(<span class="default"><a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span>
 
<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/>
<span class="kw">thus </span><a NAME="E4:84"/>
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  implies  not <font color="Maroon" title="c1">r</font> is   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</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"> ( <font color="Maroon" title="c1">r</font> is  not   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a> implies  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="84_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:84_1"/><span class="lab"><font color="Green" title="E68">A3</font></span>: 
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp; <font color="Maroon" title="c1">r</font> is   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</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">reconsider </span><font color="Maroon" title="c2">r</font> = <font color="Maroon" title="c1">r</font> as   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a> ;<br/>
<a NAME="E3:84_1"/>
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r</font></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T37" target="_self" onmouseover="rs('surrealn/T37')" onmouseout="rh()">Th37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:84_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T46" target="_self" onmouseover="rs('surrealn/T46')" onmouseout="rh()">Th46</a>, <a class="txt" href="#E1:84_1"><span class="lab"><font color="Green" title="E68">A3</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/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:84"/><span class="lab"><font color="Green" title="E68">A4</font></span>: 
<font color="Maroon" title="c1">r</font> is  not   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a>
 <span class="kw">and </span><br/><a NAME="E6:84"/><span class="lab"><font color="Green" title="E69">A5</font></span>: 
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</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/>

<a NAME="E7:84"/>
(  <a href="surrealo.html#K2" title="SURREALO:func.2">born_eq</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</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#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</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#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  )
 
<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="surrealn.html#T49" target="_self" onmouseover="rs('surrealn/T49')" onmouseout="rh()">Th49</a></span>;<br/>
<a NAME="E8:84"/><span class="kw">then </span>
 <a href="surrealo.html#K2" title="SURREALO:func.2">born_eq</a> <span class="p1">(<span class="default"><a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E3:84"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="xboole_0.html#D8" onmouseover="rs('xboole_0/D8')" onmouseout="rh()">XBOOLE_0:def 8</a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E9:84"/><span class="kw">then </span>
 <a href="surrealo.html#K2" title="SURREALO:func.2">born_eq</a> <span class="p1">(<span class="default"><a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T11" onmouseover="rs('ordinal1/T11')" onmouseout="rh()">ORDINAL1:11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">B</font> =  <a href="surrealo.html#K2" title="SURREALO:func.2">born_eq</a> <span class="p1">(<span class="default"><a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">x</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a><span class="kw"> such that </span><br/><a NAME="E12:84"/><span class="lab"><font color="Green" title="E70">A6</font></span>: 
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> &amp; <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c3">x</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></span>;<br/>
<a NAME="E13:84"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:84"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="ref" href="surreal0.html#D18" onmouseover="rs('surreal0/D18')" onmouseout="rh()">SURREAL0:def 18</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">d</font> being   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a><span class="kw"> such that </span><br/><a NAME="E15:84"/><span class="lab"><font color="Green" title="E71">A7</font></span>: 
( <font color="Maroon" title="c3">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> &amp; <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c2">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T35" target="_self" onmouseover="rs('surrealn/T35')" onmouseout="rh()">Th35</a></span>;<br/>
<a NAME="E16:84"/>
( <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> &amp; <a href="surrealn.html#K4" title="SURREALN:func.4">uDyadic</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:84"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E12:84"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a>, <a class="ref" href="surrealn.html#T46" target="_self" onmouseover="rs('surrealn/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E17:84"/><span class="kw">then </span>
( <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a>, <a class="ref" href="surrealn.html#T51" target="_self" onmouseover="rs('surrealn/T51')" onmouseout="rh()">Th51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:84"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a>, <a class="txt" href="#E5:84"><span class="lab"><font color="Green" title="E68">A4</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/>


</div>
