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<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</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">x</font> is   <a href="surrealo.html#NM1" title="SURREALO:NM.1">uSurreal</a> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) iff  ex <font color="Olive" title="b1">d</font> being   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="Olive" title="b1">d</font> )</span><br/>

<div><span class="kw">hereby </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"> (  ex <font color="Olive" title="b1">d</font> being   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="Olive" title="b1">d</font> implies ( <font color="Maroon" title="c1">x</font> is   <a href="surrealo.html#NM1" title="SURREALO:NM.1">uSurreal</a> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:59_1"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
( <font color="Maroon" title="c1">x</font> is   <a href="surrealo.html#NM1" title="SURREALO:NM.1">uSurreal</a> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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">  ex <font color="Olive" title="b1">d</font> being   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="Olive" title="b1">d</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">b</font> =  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><a NAME="E3:59_1"/>
<font color="Maroon" title="c1">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="ref" href="surreal0.html#D18" onmouseover="rs('surreal0/D18')" onmouseout="rh()">SURREAL0:def 18</a></span>;<br/><a NAME="E4:59_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">d</font> being   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a> st <br/>( <font color="Maroon" title="c1">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="Olive" title="b1">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="Olive" title="b1">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/><span class="kw">hence </span><a NAME="E5:59_1"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="Olive" title="b1">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="surrealo.html#T50" onmouseover="rs('surrealo/T50')" onmouseout="rh()">SURREALO:50</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">given </span><font color="Maroon" title="c2">d</font> being   <a href="surrealn.html#NM1" title="SURREALN:NM.1">Dyadic</a><span class="kw"> such that </span><a NAME="E3:59"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">d</font>
 ; <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">x</font> is   <a href="surrealo.html#NM1" title="SURREALO:NM.1">uSurreal</a> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:59"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<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">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="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T36" target="_self" onmouseover="rs('surrealn/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<a NAME="E6:59"/>
 <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">d</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E47">A3</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">hence </span><a NAME="E7:59"/>
( <font color="Maroon" title="c1">x</font> is   <a href="surrealo.html#NM1" title="SURREALO:NM.1">uSurreal</a> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E46">A2</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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