<?xml version="1.0"?>
<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#K5" title="SURREALI:func.5">divL</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#K5" title="SURREALI:func.5">divL</a> (<font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">I2</font>) &amp;  <a href="surreali.html#K6" title="SURREALI:func.6">divR</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#K6" title="SURREALI:func.6">divR</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#K5" title="SURREALI:func.5">divL</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#K5" title="SURREALI:func.5">divL</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">I2</font>) &amp;  <a href="surreali.html#K6" title="SURREALI:func.6">divR</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#K6" title="SURREALI:func.6">divR</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#K5" title="SURREALI:func.5">divL</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#K5" title="SURREALI:func.5">divL</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>) &amp;  <a href="surreali.html#K6" title="SURREALI:func.6">divR</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#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:24"/><span class="lab"><font color="Green" title="E22">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#K5" title="SURREALI:func.5">divL</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#K5" title="SURREALI:func.5">divL</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>) &amp;  <a href="surreali.html#K6" title="SURREALI:func.6">divR</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#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>) )</span><br/>

<a NAME="E2:24"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 <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="#E1:24"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="surreali.html#T16" target="_self" onmouseover="rs('surreali/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<a NAME="E3:24"/><span class="lab"><font color="Green" title="E24">A3</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#K5" title="SURREALI:func.5">divL</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#K5" title="SURREALI:func.5">divL</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#D5" target="_self" onmouseover="rs('surreali/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<a NAME="E4:24"/>
 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#K5" title="SURREALI:func.5">divL</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#K5" title="SURREALI:func.5">divL</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#K5" title="SURREALI:func.5">divL</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>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">o</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"> ( <font color="Maroon" title="c5">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#K5" title="SURREALI:func.5">divL</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</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="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</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="c5">o</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1"/>
<font color="Maroon" title="c5">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#K5" title="SURREALI:func.5">divL</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</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"> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</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="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</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="c5">o</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c5">o</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="#E3:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</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="c5">o</font> = 
<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> <font color="Maroon" title="c6">n</font></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#D5" target="_self" onmouseover="rs('surreali/D5')" onmouseout="rh()">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</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="c5">o</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="surreali.html#D5" target="_self" onmouseover="rs('surreali/D5')" onmouseout="rh()">Def5</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="E5:24"/>
 <a href="surreali.html#K5" title="SURREALI:func.5">divL</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#K5" title="SURREALI:func.5">divL</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="#E3:24"><span class="lab"><font color="Green" title="E24">A3</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">  <a href="surreali.html#K6" title="SURREALI:func.6">divR</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#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">I2</font>)</span><br/>

<a NAME="E6:24"/><span class="lab"><font color="Green" title="E25">A4</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#K6" title="SURREALI:func.6">divR</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#K6" title="SURREALI:func.6">divR</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#D6" target="_self" onmouseover="rs('surreali/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E7:24"/>
 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#K6" title="SURREALI:func.6">divR</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#K6" title="SURREALI:func.6">divR</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#K6" title="SURREALI:func.6">divR</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>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">o</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"> ( <font color="Maroon" title="c5">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#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</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="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</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="c5">o</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_2"/>
<font color="Maroon" title="c5">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#K6" title="SURREALI:func.6">divR</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">I1</font>)</span>)</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"> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</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="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</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="c5">o</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c5">o</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="#E6:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</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="c5">o</font> = 
<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> <font color="Maroon" title="c6">n</font></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#D6" target="_self" onmouseover="rs('surreali/D6')" onmouseout="rh()">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</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="c5">o</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D6" target="_self" onmouseover="rs('surreali/D6')" onmouseout="rh()">Def6</a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E23">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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:24"/>
 <a href="surreali.html#K6" title="SURREALI:func.6">divR</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#K6" title="SURREALI:func.6">divR</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="#E6:24"><span class="lab"><font color="Green" title="E25">A4</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/>


</div>
