<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &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>  &amp;  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> implies  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c2">y</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:25"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &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>  &amp;  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</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">  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c2">y</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">a</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E21">A1</font></span></a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</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="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E3:25"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<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="c3">a</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="E4:25"/>
 <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T11" onmouseover="rs('surrealo/T11')" onmouseout="rh()">SURREALO:11</a></span>;<br/>
<a NAME="E5:25"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="surrealn.html#T2" target="_self" onmouseover="rs('surrealn/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="surrealo.html#T17" onmouseover="rs('surrealo/T17')" onmouseout="rh()">SURREALO:17</a></span>;<br/>
<a NAME="E6:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A3</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:25"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>] &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" onmouseover="rs('nat_1/S5')" onmouseout="rh()">NAT_1:sch 5</a>(<a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E23">A3</font></span></a>);<br/></span>
<a NAME="E9:25"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:25_1"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="E2:25_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D1" target="_self" onmouseover="rs('surrealn/D1')" onmouseout="rh()">Def1</a>, <a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E8:25"><span class="lab"><font color="Green" title="E24">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:25_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="surrealo.html#T21" onmouseover="rs('surrealo/T21')" onmouseout="rh()">SURREALO:21</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">k1</font> = <font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" onmouseover="rs('nat_1/T20')" onmouseout="rh()">NAT_1:20</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">k1</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> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k1</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c2">y</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font> )</span><br/>

<a NAME="E11:25"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</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="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</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="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</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> <font color="Maroon" title="c1">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:25"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E21">A1</font></span></a></span>;<br/>
<a NAME="E12:25"/>
 for <font color="Olive" title="b1">z</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">z</font></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"><font color="Olive" title="b1">z</font></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> <font color="Maroon" title="c1">x</font> holds <br/> <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</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> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">z</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"> (  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">z</font></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"><font color="Maroon" title="c6">z</font></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> <font color="Maroon" title="c1">x</font> implies  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</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> <font color="Maroon" title="c6">z</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:25_2"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">z</font></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"><font color="Maroon" title="c6">z</font></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> <font color="Maroon" title="c1">x</font> )
 <span class="kw">and </span><br/><a NAME="E2:25_2"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
 not  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</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> <font color="Maroon" title="c6">z</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"> contradiction</span><br/>

<a NAME="E3:25_2"/><span class="lab"><font color="Green" title="E28">A8</font></span>: 
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c6">z</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="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</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#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c4">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T4" target="_self" onmouseover="rs('surrealn/T4')" onmouseout="rh()">Th4</a>, <a class="txt" href="#E2:25_2"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="ordinal1.html#T16" onmouseover="rs('ordinal1/T16')" onmouseout="rh()">ORDINAL1:16</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">bz</font> =  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c6">z</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<a NAME="E5:25_2"/>
( <font color="Maroon" title="c7">bz</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="nat_1.html#T44" onmouseover="rs('nat_1/T44')" onmouseout="rh()">NAT_1:44</a></span>;<br/>
<a NAME="E6:25_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">bz</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E7:25_2"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c7">bz</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c5">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" onmouseover="rs('nat_1/T39')" onmouseout="rh()">NAT_1:39</a></span>;<br/>
<a NAME="E8:25_2"/><span class="kw">then </span>
( <font color="Maroon" title="c6">z</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="c7">bz</font> &amp;  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c7">bz</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c5">k1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#T35" onmouseover="rs('surreal0/T35')" onmouseout="rh()">SURREAL0:35</a>, <a class="ref" href="surreal0.html#D18" onmouseover="rs('surreal0/D18')" onmouseout="rh()">SURREAL0:def 18</a></span>;<br/>
<a NAME="E9:25_2"/><span class="kw">then </span>
 <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</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="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T2" target="_self" onmouseover="rs('surrealn/T2')" onmouseout="rh()">Th2</a>, <a class="txt" href="#E1:25_2"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="surrealo.html#T17" onmouseover="rs('surrealo/T17')" onmouseout="rh()">SURREALO:17</a></span>;<br/>
<a NAME="E10:25_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:25"><span class="lab"><font color="Green" title="E24">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:25_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</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="E13:25"/>
<font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="surrealo.html#T16" onmouseover="rs('surrealo/T16')" onmouseout="rh()">SURREALO:16</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"> ( <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k1</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c2">y</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font> )</span><br/>

<a NAME="E14:25"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
<a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k1</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:25_3"/>
<font color="Maroon" title="c2">y</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k1</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"> contradiction</span><br/>

<a NAME="E2:25_3"/><span class="kw">then </span>
 <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</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="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T21" onmouseover="rs('surrealo/T21')" onmouseout="rh()">SURREALO:21</a>, <a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E21">A1</font></span></a></span>;<br/>
<a NAME="E3:25_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:25"><span class="lab"><font color="Green" title="E24">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:25_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</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="E15:25"/>
( <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k1</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c2">y</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T21" onmouseover="rs('surrealo/T21')" onmouseout="rh()">SURREALO:21</a>, <a class="txt" href="#E8:25"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E21">A1</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"> <font color="Maroon" title="c5">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span><br/>

<a NAME="E16:25"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
<font color="Maroon" title="c5">k1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c2">y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:25_4"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
 not <font color="Maroon" title="c5">k1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c2">y</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"> contradiction</span><br/>

<a NAME="E2:25_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">k2</font> = <font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" onmouseover="rs('nat_1/T20')" onmouseout="rh()">NAT_1:20</a></span>;<br/>
<a NAME="E4:25_4"/>
<font color="Maroon" title="c5">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
;<br/>
<a NAME="E5:25_4"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c5">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c6">k2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T38" onmouseover="rs('nat_1/T38')" onmouseout="rh()">NAT_1:38</a></span>;<br/>
<a NAME="E6:25_4"/><span class="kw">then </span>
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c2">y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">k2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_4"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="ref" href="ordinal1.html#T16" onmouseover="rs('ordinal1/T16')" onmouseout="rh()">ORDINAL1:16</a>, <a class="ref" href="ordinal1.html#T22" onmouseover="rs('ordinal1/T22')" onmouseout="rh()">ORDINAL1:22</a></span>;<br/>
<a NAME="E7:25_4"/><span class="kw">then </span>
( <font color="Maroon" title="c2">y</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="c2">y</font></span>)</span> &amp;  <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="c2">y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c6">k2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#T35" onmouseover="rs('surreal0/T35')" onmouseout="rh()">SURREAL0:35</a>, <a class="ref" href="surreal0.html#D18" onmouseover="rs('surreal0/D18')" onmouseout="rh()">SURREAL0:def 18</a></span>;<br/>
<a NAME="E8:25_4"/><span class="kw">then </span>
( <font color="Maroon" title="c2">y</font> <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k2</font> &amp; <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T2" target="_self" onmouseover="rs('surrealn/T2')" onmouseout="rh()">Th2</a>, <a class="txt" href="surrealn.html#E3"><span class="lab"><font color="Green" title="E3">Lm1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:25_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:25"><span class="lab"><font color="Green" title="E26">A9</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>
<a NAME="E17:25"/>
<font color="Maroon" title="c2">y</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> <font color="Maroon" title="c1">x</font></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> <font color="Maroon" title="c1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:25"/>
<font color="Maroon" title="c5">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a>, <a class="txt" href="#E16:25"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="ordinal1.html#T12" onmouseover="rs('ordinal1/T12')" onmouseout="rh()">ORDINAL1:12</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>
