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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">Z</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  for <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font> &amp; <font color="Olive" title="b1">I1</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">I2</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c1">Z</font> holds <br/> <a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">I1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">I2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">I1</font>, <font color="Maroon" title="c3">I2</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font> &amp; <font color="Maroon" title="c2">I1</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I2</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c1">Z</font> holds <br/> <a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">I1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">I2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font> &amp; <font color="Maroon" title="c2">I1</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I2</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c1">Z</font> implies  <a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>) )</span><br/>







<span class="kw">assume </span><a NAME="E1:23"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font> &amp; <font color="Maroon" title="c2">I1</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I2</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c1">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">  <a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">T1</font> =  <a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">T2</font> =  <a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>);<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><span class="p1">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E2:23"/>
<span class="p1">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D4" target="_self" onmouseover="rs('surreali/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E3:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D4" target="_self" onmouseover="rs('surreali/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E4:23"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
( <span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> &amp; <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T33" onmouseover="rs('xboole_1/T33')" onmouseout="rh()">XBOOLE_1:33</a></span>;<br/>
<a NAME="E5:23"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
 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">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</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">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:23_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
( <span class="p1">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is  <a href="xtuple_0.html#V1" title="XTUPLE_0:attr.1">pair</a>  &amp; <span class="p1">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is  <a href="xtuple_0.html#V1" title="XTUPLE_0:attr.1">pair</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D4" target="_self" onmouseover="rs('surreali/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="lab"><font color="Green" title="E27">A7</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font>,<span class="p3">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c2">I1</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="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c2">I1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D4" target="_self" onmouseover="rs('surreali/D4')" onmouseout="rh()">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font>,<span class="p3">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c3">I2</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="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c2">I1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="surreali.html#T15" target="_self" onmouseover="rs('surreali/T15')" onmouseout="rh()">Th15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font>,<span class="p3">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c3">I2</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="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c3">I2</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="surreali.html#T15" target="_self" onmouseover="rs('surreali/T15')" onmouseout="rh()">Th15</a>, <a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D4" target="_self" onmouseover="rs('surreali/D4')" onmouseout="rh()">Def4</a></span>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font>,<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c2">I1</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="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c2">I1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D4" target="_self" onmouseover="rs('surreali/D4')" onmouseout="rh()">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font>,<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c2">I1</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="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c3">I2</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="surreali.html#T15" target="_self" onmouseover="rs('surreali/T15')" onmouseout="rh()">Th15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font>,<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c3">I2</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="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c3">I2</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="surreali.html#T15" target="_self" onmouseover="rs('surreali/T15')" onmouseout="rh()">Th15</a>, <a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D4" target="_self" onmouseover="rs('surreali/D4')" onmouseout="rh()">Def4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:23_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E3:23_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="xtuple_0.html#T2" onmouseover="rs('xtuple_0/T2')" onmouseout="rh()">XTUPLE_0:2</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="E6:23"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E24">A4</font></span></a>);<br/></span>
<a NAME="E7:23"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D4" target="_self" onmouseover="rs('surreali/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E8:23"/><span class="kw">then </span>
 for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E25">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:23"/>
 <a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K4" title="SURREALI:func.4">transitions_of</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</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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