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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>] <span class="kw">means </span> for <font color="Olive" title="b1">x</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  holds <br/>(  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Olive" title="b1">x</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  &amp; (  for <font color="Olive" title="b2">y</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Olive" title="b1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b2">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  ) );<br/>
<a NAME="E1:48"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
 for <font color="Olive" title="b1">D</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st (  for <font color="Olive" title="b2">C</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b2">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">D</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">C</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">D</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">D</font> be   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</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">C</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b1">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">D</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">D</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
 for <font color="Olive" title="b1">C</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b1">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">D</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</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="c1">D</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> &amp; <font color="Maroon" title="c2">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  implies (  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c2">x</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  &amp; (  for <font color="Olive" title="b1">y</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c2">x</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b1">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:48_1"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> &amp; <font color="Maroon" title="c2">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</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="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c2">x</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  &amp; (  for <font color="Olive" title="b1">y</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c2">x</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b1">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">Nx</font> = <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">Inv</font> =  <a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>;<br/>
<a NAME="E3:48_1"/><span class="lab"><font color="Green" title="E44">A4</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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c2">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="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surreali.html#T20" target="_self" onmouseover="rs('surreali/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E4:48_1"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
 <a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> is <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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="surreali.html#V1" title="SURREALI:attr.1">-surreal-valued</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_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> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="surreali.html#D7" target="_self" onmouseover="rs('surreali/D7')" onmouseout="rh()">SURREALI:def 7</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> <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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> implies <span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1_1"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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 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#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">o</font> = <font color="Maroon" title="c5">o</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a></span>;<br/>
<a NAME="E3:48_1_1"/>
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c6">o</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="c2">x</font> &amp; <font color="Maroon" title="c6">o</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E3:48_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a>, <a class="ref" href="surreali.html#T21" target="_self" onmouseover="rs('surreali/T21')" onmouseout="rh()">Th21</a></span>;<br/>
<a NAME="E4:48_1_1"/><span class="kw">then </span>
 <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c6">o</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:48_1_1"/>
<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</font> is   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</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="E5:48_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A7</font></span>: 
(  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> is  <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T10" target="_self" onmouseover="rs('surreali/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( (  for <font color="Olive" title="b1">yL</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 holds <br/><span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b1">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  ) &amp; (  for <font color="Olive" title="b1">yR</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">yR</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 holds <br/> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b1">yR</font> ) );<br/>
<a NAME="E6:48_1"/><span class="lab"><font color="Green" title="E47">A8</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <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="48_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:48_1_2"/>
 for <font color="Olive" title="b1">yL</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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> holds <br/><span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b1">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</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">yR</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">yR</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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> holds <br/> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b1">yR</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_1_2_1"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
 <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 
;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">yL</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="c5">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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> implies <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c5">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:48_1_2_1"/><span class="lab"><font color="Green" title="E49">A10</font></span>: 
<font color="Maroon" title="c5">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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 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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c5">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> </span><br/>

<a NAME="E3:48_1_2_1"/>
<span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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> <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="surreali.html#T1" target="_self" onmouseover="rs('surreali/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E4:48_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">yL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_2_1"><span class="lab"><font color="Green" title="E49">A10</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="E5:48_1_2_1"/>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c5">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2_1"><span class="lab"><font color="Green" title="E48">A9</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">thus </span><a NAME="E2:48_1_2"/>
 for <font color="Olive" title="b1">yR</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">yR</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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> holds <br/> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b1">yR</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T1" target="_self" onmouseover="rs('surreali/T1')" onmouseout="rh()">Th1</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="E7:48_1"/><span class="lab"><font color="Green" title="E48">A11</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>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</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="48_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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>2</sub></font>[<font color="Maroon" title="c5">n</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1_3"/><span class="lab"><font color="Green" title="E49">A12</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">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>2</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">n1</font> = <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<a NAME="E2:48_1_3"/><span class="lab"><font color="Green" title="E50">A13</font></span>: 
( <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#T1" onmouseover="rs('abcmiz_1/T1')" onmouseout="rh()">ABCMIZ_1:1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:48_1_3"/>
 for <font color="Olive" title="b1">yL</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b1">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</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">yR</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">yR</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b1">yR</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">yL</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="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1_3_1"/><span class="lab"><font color="Green" title="E51">A14</font></span>: 
<font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> </span><br/>

<a NAME="E2:48_1_3_1"/>
<span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></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"><span class="p4">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p5">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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"><span class="p3">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T6" target="_self" onmouseover="rs('surreali/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E3:48_1_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></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"><span class="p3">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> or <font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_3_1"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_3_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> or <font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) or <font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_3_1_1_1"/>
<font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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"> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:48_1_3_1_1_1"/>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_3"><span class="lab"><font color="Green" title="E49">A12</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_3_1_1_2"/>
<font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">yL1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:48_1_3_1_1_2"/><span class="lab"><font color="Green" title="E52">A15</font></span>: 
( <font color="Maroon" title="c8">yL1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c8">yL1</font>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">yL1</font> = <font color="Maroon" title="c8">yL1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E3:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E2:48_1_3"><span class="lab"><font color="Green" title="E50">A13</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">xR</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:48_1_3_1_1_2"/><span class="lab"><font color="Green" title="E53">A16</font></span>: 
( <font color="Maroon" title="c10">xR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <font color="Maroon" title="c10">xR</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">and </span><br/><a NAME="E7:48_1_3_1_1_2"/><span class="lab"><font color="Green" title="E54">A17</font></span>: 
<font color="Maroon" title="c7">yL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xR</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c9">yL1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">xR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">xR</font> = <font color="Maroon" title="c10">xR</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a></span>;<br/><a NAME="E9:48_1_3_1_1_2"/><span class="lab"><font color="Green" title="E55">A18</font></span>: 
<font color="Maroon" title="c11">xR</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E10:48_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A19</font></span>: 
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c11">xR</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a>, <a class="ref" href="surreali.html#T22" target="_self" onmouseover="rs('surreali/T22')" onmouseout="rh()">Th22</a></span>;<br/><a NAME="E11:48_1_3_1_1_2"/><span class="lab"><font color="Green" title="E57">A20</font></span>: 
<font color="Maroon" title="c11">xR</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E6:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">IxR</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c11">xR</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E10:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A19</font></span></a></span>;<br/><a NAME="E13:48_1_3_1_1_2"/><span class="lab"><font color="Green" title="E58">A21</font></span>: 
<font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E10:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A19</font></span></a>, <a class="txt" href="#E11:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E57">A20</font></span></a></span>;<br/><a NAME="E14:48_1_3_1_1_2"/>
( <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</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>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E15:48_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A22</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="ref" href="surrealr.html#T32" onmouseover="rs('surrealr/T32')" onmouseout="rh()">SURREALR:32</a></span>;<br/><a NAME="E16:48_1_3_1_1_2"/>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T39" onmouseover="rs('surrealr/T39')" onmouseout="rh()">SURREALR:39</a></span>;<br/><a NAME="E17:48_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A23</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E59">A22</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E18:48_1_3_1_1_2"/><span class="lab"><font color="Green" title="E61">A24</font></span>: 
( <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> &amp; <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_3"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E3:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E17:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E60">A23</font></span></a>, <a class="ref" href="surrealr.html#T70" onmouseover="rs('surrealr/T70')" onmouseout="rh()">SURREALR:70</a></span>;<br/><a NAME="E19:48_1_3_1_1_2"/>
<font color="Maroon" title="c11">xR</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="txt" href="#E9:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E55">A18</font></span></a>, <a class="txt" href="#E6:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E20:48_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A25</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E54">A17</font></span></a>, <a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E21:48_1_3_1_1_2"/><span class="lab"><font color="Green" title="E63">A26</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a>, <a class="ref" href="surrealr.html#T67" onmouseover="rs('surrealr/T67')" onmouseout="rh()">SURREALR:67</a></span>;<br/><a NAME="E22:48_1_3_1_1_2"/>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T67" onmouseover="rs('surrealr/T67')" onmouseout="rh()">SURREALR:67</a></span>;<br/><a NAME="E23:48_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A27</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E63">A26</font></span></a>, <a class="txt" href="#E20:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E62">A25</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E24:48_1_3_1_1_2"/><span class="lab"><font color="Green" title="E65">A28</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a>, <a class="ref" href="surreali.html#D8" target="_self" onmouseover="rs('surreali/D8')" onmouseout="rh()">Def8</a>, <a class="txt" href="#E13:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A21</font></span></a></span>;<br/><a NAME="E25:48_1_3_1_1_2"/>
 <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="c11">xR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E57">A20</font></span></a></span>;<br/><a NAME="E26:48_1_3_1_1_2"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E65">A28</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E27:48_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A29</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E61">A24</font></span></a>, <a class="ref" href="surrealr.html#T70" onmouseover="rs('surrealr/T70')" onmouseout="rh()">SURREALR:70</a></span>;<br/><a NAME="E28:48_1_3_1_1_2"/><span class="lab"><font color="Green" title="E67">A30</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">yL1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a>, <a class="ref" href="surrealr.html#T69" onmouseover="rs('surrealr/T69')" onmouseout="rh()">SURREALR:69</a></span>;<br/><a NAME="E29:48_1_3_1_1_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">yL1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T69" onmouseover="rs('surrealr/T69')" onmouseout="rh()">SURREALR:69</a></span>;<br/><a NAME="E30:48_1_3_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">yL1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E31:48_1_3_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E67">A30</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E32:48_1_3_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E66">A29</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E33:48_1_3_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><a NAME="E34:48_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A31</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E64">A27</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E35:48_1_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T58" onmouseover="rs('surrealr/T58')" onmouseout="rh()">SURREALR:58</a></span>;<br/><a NAME="E36:48_1_3_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T67" onmouseover="rs('surrealr/T67')" onmouseout="rh()">SURREALR:67</a></span>;<br/><a NAME="E37:48_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A32</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a></span>;<br/><a NAME="E38:48_1_3_1_1_2"/><span class="lab"><font color="Green" title="E70">A33</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T39" onmouseover="rs('surrealr/T39')" onmouseout="rh()">SURREALR:39</a></span>;<br/><a NAME="E39:48_1_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T37" onmouseover="rs('surrealr/T37')" onmouseout="rh()">SURREALR:37</a></span>;<br/><a NAME="E40:48_1_3_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E70">A33</font></span></a>, <a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a></span>;<br/><a NAME="E41:48_1_3_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E69">A32</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E42:48_1_3_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E68">A31</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><span class="kw">hence </span><a NAME="E43:48_1_3_1_1_2"/>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A21</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:48_1_3_1_1_3"/>
<font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">yL1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:48_1_3_1_1_3"/><span class="lab"><font color="Green" title="E52">A34</font></span>: 
( <font color="Maroon" title="c8">yL1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c8">yL1</font>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">yL1</font> = <font color="Maroon" title="c8">yL1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E3:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E52">A34</font></span></a>, <a class="txt" href="#E2:48_1_3"><span class="lab"><font color="Green" title="E50">A13</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">xR</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:48_1_3_1_1_3"/><span class="lab"><font color="Green" title="E53">A35</font></span>: 
( <font color="Maroon" title="c10">xR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <font color="Maroon" title="c10">xR</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">and </span><br/><a NAME="E7:48_1_3_1_1_3"/><span class="lab"><font color="Green" title="E54">A36</font></span>: 
<font color="Maroon" title="c7">yL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xR</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c9">yL1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">xR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E52">A34</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">xR</font> = <font color="Maroon" title="c10">xR</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E53">A35</font></span></a>, <a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a></span>;<br/><a NAME="E9:48_1_3_1_1_3"/><span class="lab"><font color="Green" title="E55">A37</font></span>: 
<font color="Maroon" title="c11">xR</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E53">A35</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E10:48_1_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A38</font></span>: 
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c11">xR</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a>, <a class="ref" href="surreali.html#T22" target="_self" onmouseover="rs('surreali/T22')" onmouseout="rh()">Th22</a></span>;<br/><a NAME="E11:48_1_3_1_1_3"/><span class="lab"><font color="Green" title="E57">A39</font></span>: 
<font color="Maroon" title="c11">xR</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E6:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E53">A35</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">IxR</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c11">xR</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E10:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E56">A38</font></span></a></span>;<br/><a NAME="E13:48_1_3_1_1_3"/><span class="lab"><font color="Green" title="E58">A40</font></span>: 
<font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E10:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E56">A38</font></span></a>, <a class="txt" href="#E11:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E57">A39</font></span></a></span>;<br/><a NAME="E14:48_1_3_1_1_3"/>
( <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</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"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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#T11" onmouseover="rs('surrealo/T11')" onmouseout="rh()">SURREALO:11</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E15:48_1_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A41</font></span>: 
<font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E53">A35</font></span></a>, <a class="ref" href="surrealr.html#T32" onmouseover="rs('surrealr/T32')" onmouseout="rh()">SURREALR:32</a></span>;<br/><a NAME="E16:48_1_3_1_1_3"/>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T39" onmouseover="rs('surrealr/T39')" onmouseout="rh()">SURREALR:39</a></span>;<br/><a NAME="E17:48_1_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A42</font></span>: 
<font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E59">A41</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E18:48_1_3_1_1_3"/><span class="lab"><font color="Green" title="E61">A43</font></span>: 
( <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> &amp; <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_3"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E3:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E52">A34</font></span></a>, <a class="ref" href="surrealr.html#T71" onmouseover="rs('surrealr/T71')" onmouseout="rh()">SURREALR:71</a>, <a class="txt" href="#E17:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E60">A42</font></span></a></span>;<br/><a NAME="E19:48_1_3_1_1_3"/>
<font color="Maroon" title="c11">xR</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="txt" href="#E9:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E55">A37</font></span></a>, <a class="txt" href="#E6:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E53">A35</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E20:48_1_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A44</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E54">A36</font></span></a>, <a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E21:48_1_3_1_1_3"/><span class="lab"><font color="Green" title="E63">A45</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a>, <a class="ref" href="surrealr.html#T67" onmouseover="rs('surrealr/T67')" onmouseout="rh()">SURREALR:67</a></span>;<br/><a NAME="E22:48_1_3_1_1_3"/>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T67" onmouseover="rs('surrealr/T67')" onmouseout="rh()">SURREALR:67</a></span>;<br/><a NAME="E23:48_1_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A46</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E63">A45</font></span></a>, <a class="txt" href="#E20:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E62">A44</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E24:48_1_3_1_1_3"/><span class="lab"><font color="Green" title="E65">A47</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a>, <a class="ref" href="surreali.html#D8" target="_self" onmouseover="rs('surreali/D8')" onmouseout="rh()">Def8</a>, <a class="txt" href="#E13:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E58">A40</font></span></a></span>;<br/><a NAME="E25:48_1_3_1_1_3"/>
 <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="c11">xR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E57">A39</font></span></a></span>;<br/><a NAME="E26:48_1_3_1_1_3"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E65">A47</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E27:48_1_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A48</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E61">A43</font></span></a>, <a class="ref" href="surrealr.html#T70" onmouseover="rs('surrealr/T70')" onmouseout="rh()">SURREALR:70</a></span>;<br/><a NAME="E28:48_1_3_1_1_3"/><span class="lab"><font color="Green" title="E67">A49</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">yL1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a>, <a class="ref" href="surrealr.html#T69" onmouseover="rs('surrealr/T69')" onmouseout="rh()">SURREALR:69</a></span>;<br/><a NAME="E29:48_1_3_1_1_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">yL1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T69" onmouseover="rs('surrealr/T69')" onmouseout="rh()">SURREALR:69</a></span>;<br/><a NAME="E30:48_1_3_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">yL1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E31:48_1_3_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E67">A49</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E32:48_1_3_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E66">A48</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E33:48_1_3_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><a NAME="E34:48_1_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A50</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E64">A46</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E35:48_1_3_1_1_3"/>
<span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T58" onmouseover="rs('surrealr/T58')" onmouseout="rh()">SURREALR:58</a></span>;<br/><a NAME="E36:48_1_3_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T67" onmouseover="rs('surrealr/T67')" onmouseout="rh()">SURREALR:67</a></span>;<br/><a NAME="E37:48_1_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A51</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a></span>;<br/><a NAME="E38:48_1_3_1_1_3"/><span class="lab"><font color="Green" title="E70">A52</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T39" onmouseover="rs('surrealr/T39')" onmouseout="rh()">SURREALR:39</a></span>;<br/><a NAME="E39:48_1_3_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T37" onmouseover="rs('surrealr/T37')" onmouseout="rh()">SURREALR:37</a></span>;<br/><a NAME="E40:48_1_3_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E70">A52</font></span></a>, <a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a></span>;<br/><a NAME="E41:48_1_3_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E69">A51</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E42:48_1_3_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E68">A50</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><span class="kw">hence </span><a NAME="E43:48_1_3_1_1_3"/>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_3_1_1_3"><span class="lab"><font color="Green" title="E58">A40</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></div><span class="kw">end;</span></div>

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

<span class="kw">let </span><font color="Maroon" title="c7">yL</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="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:48_1_3"/><span class="lab"><font color="Green" title="E51">A53</font></span>: 
<font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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">  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font></span><br/>

<a NAME="E5:48_1_3"/>
<span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></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"><span class="p4">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p5">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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"><span class="p3">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T6" target="_self" onmouseover="rs('surreali/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E6:48_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></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"><span class="p3">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> or <font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_3"><span class="lab"><font color="Green" title="E51">A53</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_3_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_3_2"/><span class="kw">then </span>
( <font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> or <font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) or <font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_3_2_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_3_2_1"/>
<font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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">  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:48_1_3_2_1"/>
 <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_3"><span class="lab"><font color="Green" title="E49">A12</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_3_2_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_3_2_2"/>
<font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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">  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">yL1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:48_1_3_2_2"/><span class="lab"><font color="Green" title="E52">A54</font></span>: 
( <font color="Maroon" title="c8">yL1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c8">yL1</font>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">yL1</font> = <font color="Maroon" title="c8">yL1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E3:48_1_3_2_2"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="txt" href="#E2:48_1_3"><span class="lab"><font color="Green" title="E50">A13</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">xR</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:48_1_3_2_2"/><span class="lab"><font color="Green" title="E53">A55</font></span>: 
( <font color="Maroon" title="c10">xR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <font color="Maroon" title="c10">xR</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">and </span><br/><a NAME="E7:48_1_3_2_2"/><span class="lab"><font color="Green" title="E54">A56</font></span>: 
<font color="Maroon" title="c7">yL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xR</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c9">yL1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">xR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_3_2_2"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">xR</font> = <font color="Maroon" title="c10">xR</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_3_2_2"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a></span>;<br/><a NAME="E9:48_1_3_2_2"/><span class="lab"><font color="Green" title="E55">A57</font></span>: 
<font color="Maroon" title="c11">xR</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_3_2_2"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E10:48_1_3_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A58</font></span>: 
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c11">xR</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a>, <a class="ref" href="surreali.html#T22" target="_self" onmouseover="rs('surreali/T22')" onmouseout="rh()">Th22</a></span>;<br/><a NAME="E11:48_1_3_2_2"/><span class="lab"><font color="Green" title="E57">A59</font></span>: 
<font color="Maroon" title="c11">xR</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E6:48_1_3_2_2"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">IxR</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c11">xR</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E10:48_1_3_2_2"><span class="lab"><font color="Green" title="E56">A58</font></span></a></span>;<br/><a NAME="E13:48_1_3_2_2"/><span class="lab"><font color="Green" title="E58">A60</font></span>: 
<font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E10:48_1_3_2_2"><span class="lab"><font color="Green" title="E56">A58</font></span></a>, <a class="txt" href="#E11:48_1_3_2_2"><span class="lab"><font color="Green" title="E57">A59</font></span></a></span>;<br/><a NAME="E14:48_1_3_2_2"/>
( <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</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"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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#T11" onmouseover="rs('surrealo/T11')" onmouseout="rh()">SURREALO:11</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E15:48_1_3_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A61</font></span>: 
<font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_3_2_2"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="surrealr.html#T32" onmouseover="rs('surrealr/T32')" onmouseout="rh()">SURREALR:32</a></span>;<br/><a NAME="E16:48_1_3_2_2"/>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T39" onmouseover="rs('surrealr/T39')" onmouseout="rh()">SURREALR:39</a></span>;<br/><a NAME="E17:48_1_3_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A62</font></span>: 
<font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48_1_3_2_2"><span class="lab"><font color="Green" title="E59">A61</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E18:48_1_3_2_2"/><span class="lab"><font color="Green" title="E61">A63</font></span>: 
( <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> &amp; <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_3"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E3:48_1_3_2_2"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="ref" href="surrealr.html#T71" onmouseover="rs('surrealr/T71')" onmouseout="rh()">SURREALR:71</a>, <a class="txt" href="#E17:48_1_3_2_2"><span class="lab"><font color="Green" title="E60">A62</font></span></a></span>;<br/><a NAME="E19:48_1_3_2_2"/>
<font color="Maroon" title="c11">xR</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="txt" href="#E9:48_1_3_2_2"><span class="lab"><font color="Green" title="E55">A57</font></span></a>, <a class="txt" href="#E6:48_1_3_2_2"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E20:48_1_3_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A64</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_3_2_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a>, <a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E21:48_1_3_2_2"/><span class="lab"><font color="Green" title="E63">A65</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a>, <a class="ref" href="surrealr.html#T67" onmouseover="rs('surrealr/T67')" onmouseout="rh()">SURREALR:67</a></span>;<br/><a NAME="E22:48_1_3_2_2"/>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T67" onmouseover="rs('surrealr/T67')" onmouseout="rh()">SURREALR:67</a></span>;<br/><a NAME="E23:48_1_3_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A66</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_1_3_2_2"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="txt" href="#E20:48_1_3_2_2"><span class="lab"><font color="Green" title="E62">A64</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E24:48_1_3_2_2"/><span class="lab"><font color="Green" title="E65">A67</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a>, <a class="ref" href="surreali.html#D8" target="_self" onmouseover="rs('surreali/D8')" onmouseout="rh()">Def8</a>, <a class="txt" href="#E13:48_1_3_2_2"><span class="lab"><font color="Green" title="E58">A60</font></span></a></span>;<br/><a NAME="E25:48_1_3_2_2"/>
 <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="c11">xR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_3_2_2"><span class="lab"><font color="Green" title="E57">A59</font></span></a></span>;<br/><a NAME="E26:48_1_3_2_2"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48_1_3_2_2"><span class="lab"><font color="Green" title="E65">A67</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E27:48_1_3_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A68</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48_1_3_2_2"><span class="lab"><font color="Green" title="E61">A63</font></span></a>, <a class="ref" href="surrealr.html#T70" onmouseover="rs('surrealr/T70')" onmouseout="rh()">SURREALR:70</a></span>;<br/><a NAME="E28:48_1_3_2_2"/><span class="lab"><font color="Green" title="E67">A69</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">yL1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a>, <a class="ref" href="surrealr.html#T69" onmouseover="rs('surrealr/T69')" onmouseout="rh()">SURREALR:69</a></span>;<br/><a NAME="E29:48_1_3_2_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">yL1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T69" onmouseover="rs('surrealr/T69')" onmouseout="rh()">SURREALR:69</a></span>;<br/><a NAME="E30:48_1_3_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">yL1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E31:48_1_3_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:48_1_3_2_2"><span class="lab"><font color="Green" title="E67">A69</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E32:48_1_3_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:48_1_3_2_2"><span class="lab"><font color="Green" title="E66">A68</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E33:48_1_3_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><a NAME="E34:48_1_3_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A70</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:48_1_3_2_2"><span class="lab"><font color="Green" title="E64">A66</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E35:48_1_3_2_2"/>
<span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T58" onmouseover="rs('surrealr/T58')" onmouseout="rh()">SURREALR:58</a></span>;<br/><a NAME="E36:48_1_3_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T67" onmouseover="rs('surrealr/T67')" onmouseout="rh()">SURREALR:67</a></span>;<br/><a NAME="E37:48_1_3_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A71</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a></span>;<br/><a NAME="E38:48_1_3_2_2"/><span class="lab"><font color="Green" title="E70">A72</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T39" onmouseover="rs('surrealr/T39')" onmouseout="rh()">SURREALR:39</a></span>;<br/><a NAME="E39:48_1_3_2_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T37" onmouseover="rs('surrealr/T37')" onmouseout="rh()">SURREALR:37</a></span>;<br/><a NAME="E40:48_1_3_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:48_1_3_2_2"><span class="lab"><font color="Green" title="E70">A72</font></span></a>, <a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a></span>;<br/><a NAME="E41:48_1_3_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:48_1_3_2_2"><span class="lab"><font color="Green" title="E69">A71</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E42:48_1_3_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:48_1_3_2_2"><span class="lab"><font color="Green" title="E68">A70</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><span class="kw">hence </span><a NAME="E43:48_1_3_2_2"/>
 <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_3_2_2"><span class="lab"><font color="Green" title="E58">A60</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_3_2_3"><span class="kw">suppose </span></a><a NAME="E1:48_1_3_2_3"/>
<font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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">  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">yL1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:48_1_3_2_3"/><span class="lab"><font color="Green" title="E52">A73</font></span>: 
( <font color="Maroon" title="c8">yL1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c7">yL</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c8">yL1</font>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">yL1</font> = <font color="Maroon" title="c8">yL1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E3:48_1_3_2_3"><span class="lab"><font color="Green" title="E52">A73</font></span></a>, <a class="txt" href="#E2:48_1_3"><span class="lab"><font color="Green" title="E50">A13</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">xR</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:48_1_3_2_3"/><span class="lab"><font color="Green" title="E53">A74</font></span>: 
( <font color="Maroon" title="c10">xR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <font color="Maroon" title="c10">xR</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">and </span><br/><a NAME="E7:48_1_3_2_3"/><span class="lab"><font color="Green" title="E54">A75</font></span>: 
<font color="Maroon" title="c7">yL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">xR</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c9">yL1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">xR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_3_2_3"><span class="lab"><font color="Green" title="E52">A73</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">xR</font> = <font color="Maroon" title="c10">xR</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_3_2_3"><span class="lab"><font color="Green" title="E53">A74</font></span></a>, <a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a></span>;<br/><a NAME="E9:48_1_3_2_3"/><span class="lab"><font color="Green" title="E55">A76</font></span>: 
<font color="Maroon" title="c11">xR</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_3_2_3"><span class="lab"><font color="Green" title="E53">A74</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E10:48_1_3_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A77</font></span>: 
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c11">xR</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a>, <a class="ref" href="surreali.html#T22" target="_self" onmouseover="rs('surreali/T22')" onmouseout="rh()">Th22</a></span>;<br/><a NAME="E11:48_1_3_2_3"/><span class="lab"><font color="Green" title="E57">A78</font></span>: 
<font color="Maroon" title="c11">xR</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E6:48_1_3_2_3"><span class="lab"><font color="Green" title="E53">A74</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">IxR</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c11">xR</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E10:48_1_3_2_3"><span class="lab"><font color="Green" title="E56">A77</font></span></a></span>;<br/><a NAME="E13:48_1_3_2_3"/><span class="lab"><font color="Green" title="E58">A79</font></span>: 
<font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E10:48_1_3_2_3"><span class="lab"><font color="Green" title="E56">A77</font></span></a>, <a class="txt" href="#E11:48_1_3_2_3"><span class="lab"><font color="Green" title="E57">A78</font></span></a></span>;<br/><a NAME="E14:48_1_3_2_3"/>
( <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</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>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E15:48_1_3_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A80</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_3_2_3"><span class="lab"><font color="Green" title="E53">A74</font></span></a>, <a class="ref" href="surrealr.html#T32" onmouseover="rs('surrealr/T32')" onmouseout="rh()">SURREALR:32</a></span>;<br/><a NAME="E16:48_1_3_2_3"/>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T39" onmouseover="rs('surrealr/T39')" onmouseout="rh()">SURREALR:39</a></span>;<br/><a NAME="E17:48_1_3_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A81</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48_1_3_2_3"><span class="lab"><font color="Green" title="E59">A80</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E18:48_1_3_2_3"/><span class="lab"><font color="Green" title="E61">A82</font></span>: 
( <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> &amp; <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_3"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E3:48_1_3_2_3"><span class="lab"><font color="Green" title="E52">A73</font></span></a>, <a class="txt" href="#E17:48_1_3_2_3"><span class="lab"><font color="Green" title="E60">A81</font></span></a>, <a class="ref" href="surrealr.html#T70" onmouseover="rs('surrealr/T70')" onmouseout="rh()">SURREALR:70</a></span>;<br/><a NAME="E19:48_1_3_2_3"/>
<font color="Maroon" title="c11">xR</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="txt" href="#E9:48_1_3_2_3"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E6:48_1_3_2_3"><span class="lab"><font color="Green" title="E53">A74</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E20:48_1_3_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A83</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_3_2_3"><span class="lab"><font color="Green" title="E54">A75</font></span></a>, <a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E21:48_1_3_2_3"/><span class="lab"><font color="Green" title="E63">A84</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a>, <a class="ref" href="surrealr.html#T67" onmouseover="rs('surrealr/T67')" onmouseout="rh()">SURREALR:67</a></span>;<br/><a NAME="E22:48_1_3_2_3"/>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T67" onmouseover="rs('surrealr/T67')" onmouseout="rh()">SURREALR:67</a></span>;<br/><a NAME="E23:48_1_3_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A85</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_1_3_2_3"><span class="lab"><font color="Green" title="E63">A84</font></span></a>, <a class="txt" href="#E20:48_1_3_2_3"><span class="lab"><font color="Green" title="E62">A83</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E24:48_1_3_2_3"/><span class="lab"><font color="Green" title="E65">A86</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a>, <a class="ref" href="surreali.html#D8" target="_self" onmouseover="rs('surreali/D8')" onmouseout="rh()">Def8</a>, <a class="txt" href="#E13:48_1_3_2_3"><span class="lab"><font color="Green" title="E58">A79</font></span></a></span>;<br/><a NAME="E25:48_1_3_2_3"/>
 <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="c11">xR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_3_2_3"><span class="lab"><font color="Green" title="E57">A78</font></span></a></span>;<br/><a NAME="E26:48_1_3_2_3"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48_1_3_2_3"><span class="lab"><font color="Green" title="E65">A86</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E27:48_1_3_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A87</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48_1_3_2_3"><span class="lab"><font color="Green" title="E61">A82</font></span></a>, <a class="ref" href="surrealr.html#T70" onmouseover="rs('surrealr/T70')" onmouseout="rh()">SURREALR:70</a></span>;<br/><a NAME="E28:48_1_3_2_3"/><span class="lab"><font color="Green" title="E67">A88</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">yL1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T69" onmouseover="rs('surrealr/T69')" onmouseout="rh()">SURREALR:69</a>, <a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a></span>;<br/><a NAME="E29:48_1_3_2_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">yL1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T69" onmouseover="rs('surrealr/T69')" onmouseout="rh()">SURREALR:69</a></span>;<br/><a NAME="E30:48_1_3_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">yL1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E31:48_1_3_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:48_1_3_2_3"><span class="lab"><font color="Green" title="E67">A88</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E32:48_1_3_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:48_1_3_2_3"><span class="lab"><font color="Green" title="E66">A87</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E33:48_1_3_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">yL1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><a NAME="E34:48_1_3_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A89</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:48_1_3_2_3"><span class="lab"><font color="Green" title="E64">A85</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E35:48_1_3_2_3"/>
<span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T58" onmouseover="rs('surrealr/T58')" onmouseout="rh()">SURREALR:58</a></span>;<br/><a NAME="E36:48_1_3_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T67" onmouseover="rs('surrealr/T67')" onmouseout="rh()">SURREALR:67</a></span>;<br/><a NAME="E37:48_1_3_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A90</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a></span>;<br/><a NAME="E38:48_1_3_2_3"/><span class="lab"><font color="Green" title="E70">A91</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T39" onmouseover="rs('surrealr/T39')" onmouseout="rh()">SURREALR:39</a></span>;<br/><a NAME="E39:48_1_3_2_3"/>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T37" onmouseover="rs('surrealr/T37')" onmouseout="rh()">SURREALR:37</a></span>;<br/><a NAME="E40:48_1_3_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:48_1_3_2_3"><span class="lab"><font color="Green" title="E70">A91</font></span></a>, <a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a></span>;<br/><a NAME="E41:48_1_3_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:48_1_3_2_3"><span class="lab"><font color="Green" title="E69">A90</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E42:48_1_3_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">xR</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">IxR</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:48_1_3_2_3"><span class="lab"><font color="Green" title="E68">A89</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><span class="kw">hence </span><a NAME="E43:48_1_3_2_3"/>
 <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">yL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_3_2_3"><span class="lab"><font color="Green" title="E58">A79</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E8:48_1"/><span class="lab"><font color="Green" title="E49">A92</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>2</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="#E6:48_1"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E7:48_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>);<br/></span>
<a NAME="E9:48_1"/><span class="lab"><font color="Green" title="E50">A93</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D8" target="_self" onmouseover="rs('surreali/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<a NAME="E10:48_1"/><span class="lab"><font color="Green" title="E51">A94</font></span>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">l</font>, <font color="Maroon" title="c6">r</font> be   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="surreal0.html#D20" onmouseover="rs('surreal0/D20')" onmouseout="rh()">SURREAL0:def 20</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"> (  not <font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> or  not <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> or  not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1_4"/><span class="lab"><font color="Green" title="E52">A95</font></span>: 
( <font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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">  not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font></span><br/>

<a NAME="E2:48_1_4"/>
 <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> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:48_1_4"/><span class="lab"><font color="Green" title="E53">A96</font></span>: 
( <font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n1</font> &amp; (  for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_4"><span class="lab"><font color="Green" title="E52">A95</font></span></a>, <a class="ref" href="surreali.html#T27" target="_self" onmouseover="rs('surreali/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<a NAME="E5:48_1_4"/>
 <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> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">n2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:48_1_4"/><span class="lab"><font color="Green" title="E54">A97</font></span>: 
( <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n2</font> &amp; (  for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c8">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_4"><span class="lab"><font color="Green" title="E52">A95</font></span></a>, <a class="ref" href="surreali.html#T27" target="_self" onmouseover="rs('surreali/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<a NAME="E8:48_1_4"/>
<font color="Maroon" title="c8">n2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_4"><span class="lab"><font color="Green" title="E54">A97</font></span></a>, <a class="ref" href="surreali.html#T1" target="_self" onmouseover="rs('surreali/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">N2</font> = <font color="Maroon" title="c8">n2</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_4_1"/>
( <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_1"/>
<font color="Maroon" title="c7">n1</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">  not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font></span><br/><div class="add"><a NAME="E2:48_1_4_1_1"/><span class="kw">then </span>
<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="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> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T1" target="_self" onmouseover="rs('surreali/T1')" onmouseout="rh()">Th1</a></span>;<br/><a NAME="E3:48_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A98</font></span>: 
<font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_4"><span class="lab"><font color="Green" title="E53">A96</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E4:48_1_4_1_1"/>
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  &amp;  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c6">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E49">A92</font></span></a>, <a class="txt" href="#E7:48_1_4"><span class="lab"><font color="Green" title="E54">A97</font></span></a>, <a class="ref" href="surreali.html#D8" target="_self" onmouseover="rs('surreali/D8')" onmouseout="rh()">Def8</a></span>;<br/><a NAME="E5:48_1_4_1_1"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c6">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_1_4_1_1"/>
 not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_1"><span class="lab"><font color="Green" title="E55">A98</font></span></a>, <a class="txt" href="#E9:48_1"><span class="lab"><font color="Green" title="E50">A93</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n1</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">  not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">N1</font> = <font color="Maroon" title="c7">n1</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="E3:48_1_4_1_2"/><span class="lab"><font color="Green" title="E55">A99</font></span>: 
( <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#T1" onmouseover="rs('abcmiz_1/T1')" onmouseout="rh()">ABCMIZ_1:1</a></span>;<br/><a NAME="E4:48_1_4_1_2"/>
<span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">N1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font></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"><span class="p4">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p5">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font></span>)</span>,<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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"><span class="p3">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T6" target="_self" onmouseover="rs('surreali/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E5:48_1_4_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font></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"><span class="p3">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> or <font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_4"><span class="lab"><font color="Green" title="E53">A96</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_4_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font> or <font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) or <font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_1"/>
<font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</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">  not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">N1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">N1</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="#E4:48_1_4"><span class="lab"><font color="Green" title="E53">A96</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_1_4_1_2_1_1"/>
 not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</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>; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_2"/>
<font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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">  not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:48_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E56">A100</font></span>: 
( <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font> &amp; <font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c11">y1</font>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">y1</font> = <font color="Maroon" title="c11">y1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E56">A100</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2"><span class="lab"><font color="Green" title="E55">A99</font></span></a>, <a class="txt" href="#E5:48_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:48_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E57">A101</font></span>: 
( <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <font color="Maroon" title="c13">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E56">A100</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c13">x1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E57">A101</font></span></a>, <a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a></span>;<br/><a NAME="E8:48_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E58">A102</font></span>: 
( <font color="Maroon" title="c14">x1</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp; <font color="Maroon" title="c14">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a>, <a class="txt" href="#E6:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E57">A101</font></span></a></span>;<br/><a NAME="E9:48_1_4_1_2_1_2"/><span class="kw">then </span>
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c14">x1</font> &amp; <font color="Maroon" title="c14">x1</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="c2">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="c2">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E10:48_1_4_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A103</font></span>: 
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c14">x1</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="c2">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></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">Ix1</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c14">x1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E58">A102</font></span></a>, <a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/><a NAME="E12:48_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E60">A104</font></span>: 
<font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E10:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E59">A103</font></span></a>, <a class="txt" href="#E8:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E58">A102</font></span></a></span>;<br/><a NAME="E13:48_1_4_1_2_1_2"/>
<font color="Maroon" title="c14">x1</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E57">A101</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E14:48_1_4_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="txt" href="#E6:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E57">A101</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E15:48_1_4_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A105</font></span>: 
<font color="Maroon" title="c15">Ix1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E16:48_1_4_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">N2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></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"><span class="p4">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p5">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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"><span class="p3">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T6" target="_self" onmouseover="rs('surreali/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E17:48_1_4_1_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></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"><span class="p3">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> or <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_4"><span class="lab"><font color="Green" title="E54">A97</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_4_1_2_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font> or <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) or <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_2_1_1"/>
<font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</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">  not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">N2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">N2</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="#E7:48_1_4"><span class="lab"><font color="Green" title="E54">A97</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_1_4_1_2_1_2_1_1"/>
 not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</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>; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_2_1_2"/>
<font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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">  not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:48_1_4_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E62">A106</font></span>: 
( <font color="Maroon" title="c16">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font> &amp; <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c16">y2</font>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">y2</font> = <font color="Maroon" title="c16">y2</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E62">A106</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2"><span class="lab"><font color="Green" title="E55">A99</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:48_1_4_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E63">A107</font></span>: 
( <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <font color="Maroon" title="c18">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">x2</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E62">A106</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">x2</font> = <font color="Maroon" title="c18">x2</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E63">A107</font></span></a>, <a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a></span>;<br/><a NAME="E8:48_1_4_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E64">A108</font></span>: 
( <font color="Maroon" title="c19">x2</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp; <font color="Maroon" title="c19">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a>, <a class="txt" href="#E6:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E63">A107</font></span></a></span>;<br/><a NAME="E9:48_1_4_1_2_1_2_1_2"/><span class="kw">then </span>
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c19">x2</font> &amp; <font color="Maroon" title="c19">x2</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="c2">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="c2">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E10:48_1_4_1_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A109</font></span>: 
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c19">x2</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="c2">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></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">Ix2</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c19">x2</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E64">A108</font></span></a>, <a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/><a NAME="E12:48_1_4_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E66">A110</font></span>: 
<font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c20">Ix2</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E10:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E65">A109</font></span></a>, <a class="txt" href="#E8:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E64">A108</font></span></a></span>;<br/><a NAME="E13:48_1_4_1_2_1_2_1_2"/>
<font color="Maroon" title="c19">x2</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E63">A107</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E14:48_1_4_1_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A111</font></span>: 
<font color="Maroon" title="c19">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="txt" href="#E6:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E63">A107</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E15:48_1_4_1_2_1_2_1_2"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_4_1_2_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_1_4_1_2_1_2_1_2_1"/>
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</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"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="surrealo.html#T11" onmouseover="rs('surrealo/T11')" onmouseout="rh()">SURREALO:11</a></span>;<br/>
<a NAME="E2:48_1_4_1_2_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A112</font></span>: 
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp;  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a>, <a class="txt" href="#E6:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E57">A101</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E63">A107</font></span></a></span>;<br/>
<a NAME="E3:48_1_4_1_2_1_2_1_2_1"/>
 <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</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> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#T45" onmouseover="rs('surreal0/T45')" onmouseout="rh()">SURREAL0:45</a></span>;<br/>
<a NAME="E4:48_1_4_1_2_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A113</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E63">A107</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E57">A101</font></span></a>, <a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a></span>;<br/>
<a NAME="E5:48_1_4_1_2_1_2_1_2_1"/><span class="lab"><font color="Green" title="E70">A114</font></span>: 
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span> &amp;  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E56">A100</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E62">A106</font></span></a>, <a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E49">A92</font></span></a>, <a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a></span>;<br/>
<a NAME="E6:48_1_4_1_2_1_2_1_2_1"/><span class="lab"><font color="Green" title="E71">A115</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c17">y2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T29" target="_self" onmouseover="rs('surreali/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E7:48_1_4_1_2_1_2_1_2_1"/><span class="lab"><font color="Green" title="E72">A116</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c12">y1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T29" target="_self" onmouseover="rs('surreali/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E8:48_1_4_1_2_1_2_1_2_1"/><span class="lab"><font color="Green" title="E73">A117</font></span>: 
( <font color="Maroon" title="c19">x2</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp; <font color="Maroon" title="c14">x1</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E63">A107</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E57">A101</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E9:48_1_4_1_2_1_2_1_2_1"/><span class="lab"><font color="Green" title="E74">A118</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_4_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E69">A113</font></span></a>, <a class="txt" href="#E5:48_1_4_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E70">A114</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/>
<a NAME="E10:48_1_4_1_2_1_2_1_2_1"/><span class="lab"><font color="Green" title="E75">A119</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_4_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E69">A113</font></span></a>, <a class="txt" href="#E5:48_1_4_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E70">A114</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_4_1_2_1_2_1_2_1_1"/>
( <font color="Maroon" title="c12">y1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c17">y2</font> or <font color="Maroon" title="c12">y1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c17">y2</font> or <font color="Maroon" title="c17">y2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">y1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c12">y1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c17">y2</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="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_2_1_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c17">y2</font> &amp; <font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c17">y2</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a>, <a class="ref" href="surrealr.html#T39" onmouseover="rs('surrealr/T39')" onmouseout="rh()">SURREALR:39</a></span>;<br/><a NAME="E3:48_1_4_1_2_1_2_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E4:48_1_4_1_2_1_2_1_2_1_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font> &amp; <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a></span>;<br/><a NAME="E5:48_1_4_1_2_1_2_1_2_1_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> &amp; <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a></span>;<br/><a NAME="E6:48_1_4_1_2_1_2_1_2_1_1_1"/><span class="kw">then </span>
<a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_4_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E74">A118</font></span></a>, <a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_1_4_1_2_1_2_1_2_1_1_1"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E71">A115</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_2_1_2_1_1_2"/>
<font color="Maroon" title="c12">y1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c17">y2</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="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_2_1_2_1_1_2"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c12">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a></span>;<br/><a NAME="E3:48_1_4_1_2_1_2_1_2_1_1_2"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E73">A117</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E4:48_1_4_1_2_1_2_1_2_1_1_2"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_4_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E68">A112</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E5:48_1_4_1_2_1_2_1_2_1_1_2"/><span class="kw">then </span>
<a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1_4_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E75">A119</font></span></a>, <a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_1_4_1_2_1_2_1_2_1_1_2"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_4_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E72">A116</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_2_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_2_1_2_1_1_3"/>
<font color="Maroon" title="c17">y2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">y1</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="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_2_1_2_1_1_3"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c17">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a></span>;<br/><a NAME="E3:48_1_4_1_2_1_2_1_2_1_1_3"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E73">A117</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E4:48_1_4_1_2_1_2_1_2_1_1_3"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_4_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E68">A112</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E5:48_1_4_1_2_1_2_1_2_1_1_3"/><span class="kw">then </span>
<a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_4_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E74">A118</font></span></a>, <a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_1_4_1_2_1_2_1_2_1_1_3"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E71">A115</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E16:48_1_4_1_2_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a></span>;<br/><a NAME="E17:48_1_4_1_2_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c20">Ix2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E64">A108</font></span></a>, <a class="txt" href="#E8:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E58">A102</font></span></a>, <a class="txt" href="#E12:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E60">A104</font></span></a>, <a class="txt" href="#E12:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E66">A110</font></span></a>, <a class="ref" href="surreali.html#T28" target="_self" onmouseover="rs('surreali/T28')" onmouseout="rh()">Th28</a></span>;<br/><span class="kw">hence </span><a NAME="E18:48_1_4_1_2_1_2_1_2"/>
 not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E57">A101</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E63">A107</font></span></a>, <a class="txt" href="#E15:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E61">A105</font></span></a>, <a class="txt" href="#E14:48_1_4_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E67">A111</font></span></a>, <a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_2_1_3"/>
<font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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">  not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:48_1_4_1_2_1_2_1_3"/><span class="lab"><font color="Green" title="E62">A120</font></span>: 
( <font color="Maroon" title="c16">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font> &amp; <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c16">y2</font>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">y2</font> = <font color="Maroon" title="c16">y2</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E62">A120</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2"><span class="lab"><font color="Green" title="E55">A99</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:48_1_4_1_2_1_2_1_3"/><span class="lab"><font color="Green" title="E63">A121</font></span>: 
( <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <font color="Maroon" title="c18">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">x2</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E62">A120</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">x2</font> = <font color="Maroon" title="c18">x2</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E63">A121</font></span></a>, <a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a></span>;<br/><a NAME="E8:48_1_4_1_2_1_2_1_3"/><span class="lab"><font color="Green" title="E64">A122</font></span>: 
( <font color="Maroon" title="c19">x2</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp; <font color="Maroon" title="c19">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a>, <a class="txt" href="#E6:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E63">A121</font></span></a></span>;<br/><a NAME="E9:48_1_4_1_2_1_2_1_3"/><span class="kw">then </span>
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c19">x2</font> &amp; <font color="Maroon" title="c19">x2</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="c2">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="c2">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E10:48_1_4_1_2_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A123</font></span>: 
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c19">x2</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="c2">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></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">Ix2</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c19">x2</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E64">A122</font></span></a>, <a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/><a NAME="E12:48_1_4_1_2_1_2_1_3"/><span class="lab"><font color="Green" title="E66">A124</font></span>: 
<font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c20">Ix2</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E10:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E65">A123</font></span></a>, <a class="txt" href="#E8:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E64">A122</font></span></a></span>;<br/><a NAME="E13:48_1_4_1_2_1_2_1_3"/>
<font color="Maroon" title="c19">x2</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E63">A121</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E14:48_1_4_1_2_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A125</font></span>: 
<font color="Maroon" title="c19">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="txt" href="#E6:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E63">A121</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E15:48_1_4_1_2_1_2_1_3"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_4_1_2_1_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_1_4_1_2_1_2_1_3_1"/>
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</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"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="surrealo.html#T11" onmouseover="rs('surrealo/T11')" onmouseout="rh()">SURREALO:11</a></span>;<br/>
<a NAME="E2:48_1_4_1_2_1_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A126</font></span>: 
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E57">A101</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E63">A121</font></span></a>, <a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a>, <a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/>
<a NAME="E3:48_1_4_1_2_1_2_1_3_1"/>
( <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  &amp;  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E56">A100</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E62">A120</font></span></a>, <a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E49">A92</font></span></a></span>;<br/>
<a NAME="E4:48_1_4_1_2_1_2_1_3_1"/><span class="kw">then </span>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<a NAME="E5:48_1_4_1_2_1_2_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">y1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c17">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D8" target="_self" onmouseover="rs('surreali/D8')" onmouseout="rh()">Def8</a>, <a class="ref" href="surrealr.html#T73" onmouseover="rs('surrealr/T73')" onmouseout="rh()">SURREALR:73</a></span>;<br/>
<a NAME="E6:48_1_4_1_2_1_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A127</font></span>: 
( <font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c17">y2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp;  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c12">y1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a>, <a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/>
<a NAME="E7:48_1_4_1_2_1_2_1_3_1"/><span class="lab"><font color="Green" title="E70">A128</font></span>: 
<a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E62">A120</font></span></a>, <a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E49">A92</font></span></a>, <a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/>
<a NAME="E8:48_1_4_1_2_1_2_1_3_1"/><span class="lab"><font color="Green" title="E71">A129</font></span>: 
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span> &amp;  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E56">A100</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E62">A120</font></span></a>, <a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E49">A92</font></span></a>, <a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a></span>;<br/>
<a NAME="E9:48_1_4_1_2_1_2_1_3_1"/><span class="lab"><font color="Green" title="E72">A130</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c17">y2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T29" target="_self" onmouseover="rs('surreali/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E10:48_1_4_1_2_1_2_1_3_1"/><span class="lab"><font color="Green" title="E73">A131</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c12">y1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T29" target="_self" onmouseover="rs('surreali/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E11:48_1_4_1_2_1_2_1_3_1"/><span class="lab"><font color="Green" title="E74">A132</font></span>: 
( <font color="Maroon" title="c19">x2</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp; <font color="Maroon" title="c14">x1</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E63">A121</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E57">A101</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E12:48_1_4_1_2_1_2_1_3_1"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_4_1_2_1_2_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_1_4_1_2_1_2_1_3_1_1"/><span class="lab"><font color="Green" title="E75">A133</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T69" onmouseover="rs('surrealr/T69')" onmouseout="rh()">SURREALR:69</a></span>;<br/>
<a NAME="E2:48_1_4_1_2_1_2_1_3_1_1"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_2_1_3_1"><span class="lab"><font color="Green" title="E69">A127</font></span></a>, <a class="txt" href="#E2:48_1_4_1_2_1_2_1_3_1"><span class="lab"><font color="Green" title="E68">A126</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/>
<a NAME="E3:48_1_4_1_2_1_2_1_3_1_1"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_4_1_2_1_2_1_3_1"><span class="lab"><font color="Green" title="E74">A132</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/>
<a NAME="E4:48_1_4_1_2_1_2_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A134</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_4_1_2_1_2_1_3_1_1"><span class="lab"><font color="Green" title="E75">A133</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_2_1_3_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_4_1_2_1_2_1_3_1_1_1"/>
( <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c14">x1</font> or <font color="Maroon" title="c14">x1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c19">x2</font> or <font color="Maroon" title="c14">x1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c19">x2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_2_1_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_2_1_3_1_1_1_1"/>
<font color="Maroon" title="c19">x2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c14">x1</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="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_2_1_3_1_1_1_1"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a></span>;<br/><a NAME="E3:48_1_4_1_2_1_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A135</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_2_1_3_1"><span class="lab"><font color="Green" title="E71">A129</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E4:48_1_4_1_2_1_2_1_3_1_1_1_1"/>
<a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_4_1_2_1_2_1_3_1_1"><span class="lab"><font color="Green" title="E76">A134</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2_1_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E77">A135</font></span></a>, <a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><span class="kw">hence </span><a NAME="E5:48_1_4_1_2_1_2_1_3_1_1_1_1"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_4_1_2_1_2_1_3_1"><span class="lab"><font color="Green" title="E72">A130</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_2_1_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_2_1_3_1_1_1_2"/>
<font color="Maroon" title="c14">x1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c19">x2</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="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_2_1_3_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span> &amp; <font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font> &amp; <font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a>, <a class="ref" href="surrealr.html#T39" onmouseover="rs('surrealr/T39')" onmouseout="rh()">SURREALR:39</a></span>;<br/><a NAME="E3:48_1_4_1_2_1_2_1_3_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E4:48_1_4_1_2_1_2_1_3_1_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span> &amp; <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a></span>;<br/><a NAME="E5:48_1_4_1_2_1_2_1_3_1_1_1_2"/><span class="kw">then </span>
<a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_4_1_2_1_2_1_3_1_1"><span class="lab"><font color="Green" title="E76">A134</font></span></a>, <a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_1_4_1_2_1_2_1_3_1_1_1_2"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_4_1_2_1_2_1_3_1"><span class="lab"><font color="Green" title="E72">A130</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_2_1_3_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_2_1_3_1_1_1_3"/>
<font color="Maroon" title="c14">x1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c19">x2</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="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_2_1_3_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/><a NAME="E3:48_1_4_1_2_1_2_1_3_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A136</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_4_1_2_1_2_1_3_1"><span class="lab"><font color="Green" title="E70">A128</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E4:48_1_4_1_2_1_2_1_3_1_1_1_3"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_2_1_3_1"><span class="lab"><font color="Green" title="E69">A127</font></span></a>, <a class="txt" href="#E11:48_1_4_1_2_1_2_1_3_1"><span class="lab"><font color="Green" title="E74">A132</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E5:48_1_4_1_2_1_2_1_3_1_1_1_3"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_4_1_2_1_2_1_3_1"><span class="lab"><font color="Green" title="E68">A126</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E6:48_1_4_1_2_1_2_1_3_1_1_1_3"/><span class="kw">then </span>
<a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_2_1_3_1_1_1_3"><span class="lab"><font color="Green" title="E77">A136</font></span></a>, <a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_1_4_1_2_1_2_1_3_1_1_1_3"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1_4_1_2_1_2_1_3_1"><span class="lab"><font color="Green" title="E73">A131</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:48_1_4_1_2_1_2_1_3_1"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</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"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E16:48_1_4_1_2_1_2_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a></span>;<br/><a NAME="E17:48_1_4_1_2_1_2_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c20">Ix2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E64">A122</font></span></a>, <a class="txt" href="#E8:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E58">A102</font></span></a>, <a class="txt" href="#E12:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E60">A104</font></span></a>, <a class="txt" href="#E12:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E66">A124</font></span></a>, <a class="ref" href="surreali.html#T28" target="_self" onmouseover="rs('surreali/T28')" onmouseout="rh()">Th28</a></span>;<br/><span class="kw">hence </span><a NAME="E18:48_1_4_1_2_1_2_1_3"/>
 not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E57">A101</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E63">A121</font></span></a>, <a class="txt" href="#E15:48_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E61">A105</font></span></a>, <a class="txt" href="#E14:48_1_4_1_2_1_2_1_3"><span class="lab"><font color="Green" title="E67">A125</font></span></a>, <a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_3"/>
<font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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">  not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:48_1_4_1_2_1_3"/><span class="lab"><font color="Green" title="E56">A137</font></span>: 
( <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">N1</font> &amp; <font color="Maroon" title="c5">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c11">y1</font>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">y1</font> = <font color="Maroon" title="c11">y1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E56">A137</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2"><span class="lab"><font color="Green" title="E55">A99</font></span></a>, <a class="txt" href="#E5:48_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:48_1_4_1_2_1_3"/><span class="lab"><font color="Green" title="E57">A138</font></span>: 
( <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <font color="Maroon" title="c13">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E56">A137</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c13">x1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E57">A138</font></span></a>, <a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a></span>;<br/><a NAME="E8:48_1_4_1_2_1_3"/><span class="lab"><font color="Green" title="E58">A139</font></span>: 
( <font color="Maroon" title="c14">x1</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp; <font color="Maroon" title="c14">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a>, <a class="txt" href="#E6:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E57">A138</font></span></a></span>;<br/><a NAME="E9:48_1_4_1_2_1_3"/><span class="kw">then </span>
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c14">x1</font> &amp; <font color="Maroon" title="c14">x1</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="c2">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="c2">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E10:48_1_4_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A140</font></span>: 
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c14">x1</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="c2">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></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">Ix1</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c14">x1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E58">A139</font></span></a>, <a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/><a NAME="E12:48_1_4_1_2_1_3"/><span class="lab"><font color="Green" title="E60">A141</font></span>: 
<font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E10:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E59">A140</font></span></a>, <a class="txt" href="#E8:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E58">A139</font></span></a></span>;<br/><a NAME="E13:48_1_4_1_2_1_3"/>
<font color="Maroon" title="c14">x1</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E57">A138</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E14:48_1_4_1_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="txt" href="#E6:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E57">A138</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E15:48_1_4_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A142</font></span>: 
<font color="Maroon" title="c15">Ix1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E16:48_1_4_1_2_1_3"/>
<span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">N2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></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"><span class="p4">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p5">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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"><span class="p3">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T6" target="_self" onmouseover="rs('surreali/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E17:48_1_4_1_2_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></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"><span class="p3">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> or <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_4"><span class="lab"><font color="Green" title="E54">A97</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_4_1_2_1_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font> or <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) or <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_3_1_1"/>
<font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</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">  not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">N2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">N2</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="#E7:48_1_4"><span class="lab"><font color="Green" title="E54">A97</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_1_4_1_2_1_3_1_1"/>
 not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</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>; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_3_1_2"/>
<font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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">  not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:48_1_4_1_2_1_3_1_2"/><span class="lab"><font color="Green" title="E62">A143</font></span>: 
( <font color="Maroon" title="c16">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font> &amp; <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c16">y2</font>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">y2</font> = <font color="Maroon" title="c16">y2</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E62">A143</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2"><span class="lab"><font color="Green" title="E55">A99</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:48_1_4_1_2_1_3_1_2"/><span class="lab"><font color="Green" title="E63">A144</font></span>: 
( <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <font color="Maroon" title="c18">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">x2</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E62">A143</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">x2</font> = <font color="Maroon" title="c18">x2</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E63">A144</font></span></a>, <a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a></span>;<br/><a NAME="E8:48_1_4_1_2_1_3_1_2"/><span class="lab"><font color="Green" title="E64">A145</font></span>: 
( <font color="Maroon" title="c19">x2</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp; <font color="Maroon" title="c19">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a>, <a class="txt" href="#E6:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E63">A144</font></span></a></span>;<br/><a NAME="E9:48_1_4_1_2_1_3_1_2"/><span class="kw">then </span>
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c19">x2</font> &amp; <font color="Maroon" title="c19">x2</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="c2">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="c2">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E10:48_1_4_1_2_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A146</font></span>: 
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c19">x2</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="c2">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></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">Ix2</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c19">x2</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E64">A145</font></span></a>, <a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/><a NAME="E12:48_1_4_1_2_1_3_1_2"/><span class="lab"><font color="Green" title="E66">A147</font></span>: 
<font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c20">Ix2</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E10:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E65">A146</font></span></a>, <a class="txt" href="#E8:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E64">A145</font></span></a></span>;<br/><a NAME="E13:48_1_4_1_2_1_3_1_2"/>
<font color="Maroon" title="c19">x2</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E63">A144</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E14:48_1_4_1_2_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A148</font></span>: 
<font color="Maroon" title="c19">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="txt" href="#E6:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E63">A144</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E15:48_1_4_1_2_1_3_1_2"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_4_1_2_1_3_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_1_4_1_2_1_3_1_2_1"/>
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</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"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="surrealo.html#T11" onmouseover="rs('surrealo/T11')" onmouseout="rh()">SURREALO:11</a></span>;<br/>
<a NAME="E2:48_1_4_1_2_1_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A149</font></span>: 
( <font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp;  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E57">A138</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E63">A144</font></span></a>, <a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a>, <a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/>
<a NAME="E3:48_1_4_1_2_1_3_1_2_1"/>
( <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  &amp;  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E56">A137</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E62">A143</font></span></a>, <a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E49">A92</font></span></a></span>;<br/>
<a NAME="E4:48_1_4_1_2_1_3_1_2_1"/><span class="kw">then </span>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<a NAME="E5:48_1_4_1_2_1_3_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">y2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D8" target="_self" onmouseover="rs('surreali/D8')" onmouseout="rh()">Def8</a>, <a class="ref" href="surrealr.html#T73" onmouseover="rs('surrealr/T73')" onmouseout="rh()">SURREALR:73</a></span>;<br/>
<a NAME="E6:48_1_4_1_2_1_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A150</font></span>: 
( <font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c12">y1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp;  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c17">y2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a>, <a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/>
<a NAME="E7:48_1_4_1_2_1_3_1_2_1"/><span class="lab"><font color="Green" title="E70">A151</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E62">A143</font></span></a>, <a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E49">A92</font></span></a>, <a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a></span>;<br/>
<a NAME="E8:48_1_4_1_2_1_3_1_2_1"/><span class="lab"><font color="Green" title="E71">A152</font></span>: 
( <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E56">A137</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E62">A143</font></span></a>, <a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E49">A92</font></span></a>, <a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/>
<a NAME="E9:48_1_4_1_2_1_3_1_2_1"/><span class="lab"><font color="Green" title="E72">A153</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c17">y2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T29" target="_self" onmouseover="rs('surreali/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E10:48_1_4_1_2_1_3_1_2_1"/><span class="lab"><font color="Green" title="E73">A154</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c12">y1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T29" target="_self" onmouseover="rs('surreali/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E11:48_1_4_1_2_1_3_1_2_1"/><span class="lab"><font color="Green" title="E74">A155</font></span>: 
( <font color="Maroon" title="c19">x2</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp; <font color="Maroon" title="c14">x1</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E63">A144</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E57">A138</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E12:48_1_4_1_2_1_3_1_2_1"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_4_1_2_1_3_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_1_4_1_2_1_3_1_2_1_1"/><span class="lab"><font color="Green" title="E75">A156</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T69" onmouseover="rs('surrealr/T69')" onmouseout="rh()">SURREALR:69</a></span>;<br/>
<a NAME="E2:48_1_4_1_2_1_3_1_2_1_1"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E69">A150</font></span></a>, <a class="txt" href="#E2:48_1_4_1_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E68">A149</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/>
<a NAME="E3:48_1_4_1_2_1_3_1_2_1_1"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_4_1_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E74">A155</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/>
<a NAME="E4:48_1_4_1_2_1_3_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A157</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_4_1_2_1_3_1_2_1_1"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_3_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_4_1_2_1_3_1_2_1_1_1"/>
( <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c14">x1</font> or <font color="Maroon" title="c14">x1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c19">x2</font> or <font color="Maroon" title="c14">x1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c19">x2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_3_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_3_1_2_1_1_1_1"/>
<font color="Maroon" title="c19">x2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c14">x1</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="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_3_1_2_1_1_1_1"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a></span>;<br/><a NAME="E3:48_1_4_1_2_1_3_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A158</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_4_1_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E70">A151</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E4:48_1_4_1_2_1_3_1_2_1_1_1_1"/>
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> &amp; <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_4_1_2_1_3_1_2_1_1"><span class="lab"><font color="Green" title="E76">A157</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2_1_3_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E77">A158</font></span></a>, <a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><span class="kw">hence </span><a NAME="E5:48_1_4_1_2_1_3_1_2_1_1_1_1"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1_4_1_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E73">A154</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_3_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_3_1_2_1_1_1_2"/>
<font color="Maroon" title="c14">x1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c19">x2</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="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_3_1_2_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font> &amp; <font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a>, <a class="ref" href="surrealr.html#T39" onmouseover="rs('surrealr/T39')" onmouseout="rh()">SURREALR:39</a></span>;<br/><a NAME="E3:48_1_4_1_2_1_3_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E4:48_1_4_1_2_1_3_1_2_1_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span> &amp; <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a></span>;<br/><a NAME="E5:48_1_4_1_2_1_3_1_2_1_1_1_2"/><span class="kw">then </span>
<a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_4_1_2_1_3_1_2_1_1"><span class="lab"><font color="Green" title="E76">A157</font></span></a>, <a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_1_4_1_2_1_3_1_2_1_1_1_2"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1_4_1_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E73">A154</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_3_1_2_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_3_1_2_1_1_1_3"/>
<font color="Maroon" title="c14">x1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c19">x2</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="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_3_1_2_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/><a NAME="E3:48_1_4_1_2_1_3_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A159</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E71">A152</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E4:48_1_4_1_2_1_3_1_2_1_1_1_3"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E69">A150</font></span></a>, <a class="txt" href="#E11:48_1_4_1_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E74">A155</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E5:48_1_4_1_2_1_3_1_2_1_1_1_3"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_4_1_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E68">A149</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E6:48_1_4_1_2_1_3_1_2_1_1_1_3"/><span class="kw">then </span>
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> &amp; <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_3_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E77">A159</font></span></a>, <a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_1_4_1_2_1_3_1_2_1_1_1_3"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_4_1_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E72">A153</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:48_1_4_1_2_1_3_1_2_1"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</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"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E16:48_1_4_1_2_1_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a></span>;<br/><a NAME="E17:48_1_4_1_2_1_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c20">Ix2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E64">A145</font></span></a>, <a class="txt" href="#E8:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E58">A139</font></span></a>, <a class="txt" href="#E12:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E60">A141</font></span></a>, <a class="txt" href="#E12:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E66">A147</font></span></a>, <a class="ref" href="surreali.html#T28" target="_self" onmouseover="rs('surreali/T28')" onmouseout="rh()">Th28</a></span>;<br/><span class="kw">hence </span><a NAME="E18:48_1_4_1_2_1_3_1_2"/>
 not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E57">A138</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E63">A144</font></span></a>, <a class="txt" href="#E15:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E61">A142</font></span></a>, <a class="txt" href="#E14:48_1_4_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E67">A148</font></span></a>, <a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_3_1_3"/>
<font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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">  not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:48_1_4_1_2_1_3_1_3"/><span class="lab"><font color="Green" title="E62">A160</font></span>: 
( <font color="Maroon" title="c16">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N2</font> &amp; <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c16">y2</font>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">y2</font> = <font color="Maroon" title="c16">y2</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E62">A160</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2"><span class="lab"><font color="Green" title="E55">A99</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:48_1_4_1_2_1_3_1_3"/><span class="lab"><font color="Green" title="E63">A161</font></span>: 
( <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <font color="Maroon" title="c18">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">x2</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E62">A160</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">x2</font> = <font color="Maroon" title="c18">x2</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E63">A161</font></span></a>, <a class="ref" href="surreal0.html#D16" onmouseover="rs('surreal0/D16')" onmouseout="rh()">SURREAL0:def 16</a></span>;<br/><a NAME="E8:48_1_4_1_2_1_3_1_3"/><span class="lab"><font color="Green" title="E64">A162</font></span>: 
( <font color="Maroon" title="c19">x2</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp; <font color="Maroon" title="c19">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a>, <a class="txt" href="#E6:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E63">A161</font></span></a></span>;<br/><a NAME="E9:48_1_4_1_2_1_3_1_3"/><span class="kw">then </span>
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c19">x2</font> &amp; <font color="Maroon" title="c19">x2</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="c2">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="c2">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E10:48_1_4_1_2_1_3_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A163</font></span>: 
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c19">x2</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="c2">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></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">Ix2</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c19">x2</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E64">A162</font></span></a>, <a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/><a NAME="E12:48_1_4_1_2_1_3_1_3"/><span class="lab"><font color="Green" title="E66">A164</font></span>: 
<font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c20">Ix2</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E10:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E65">A163</font></span></a>, <a class="txt" href="#E8:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E64">A162</font></span></a></span>;<br/><a NAME="E13:48_1_4_1_2_1_3_1_3"/>
<font color="Maroon" title="c19">x2</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E63">A161</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E14:48_1_4_1_2_1_3_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A165</font></span>: 
<font color="Maroon" title="c19">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="txt" href="#E6:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E63">A161</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E15:48_1_4_1_2_1_3_1_3"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_4_1_2_1_3_1_3_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_1_4_1_2_1_3_1_3_1"/>
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</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"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="surrealo.html#T11" onmouseover="rs('surrealo/T11')" onmouseout="rh()">SURREALO:11</a></span>;<br/>
<a NAME="E2:48_1_4_1_2_1_3_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A166</font></span>: 
( <font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp;  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c14">x1</font> &amp; <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp;  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E57">A138</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E63">A161</font></span></a>, <a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a>, <a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/>
<a NAME="E3:48_1_4_1_2_1_3_1_3_1"/>
 <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</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> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#T45" onmouseover="rs('surreal0/T45')" onmouseout="rh()">SURREAL0:45</a></span>;<br/>
<a NAME="E4:48_1_4_1_2_1_3_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A167</font></span>: 
<font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E63">A161</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E57">A138</font></span></a>, <a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/>
<a NAME="E5:48_1_4_1_2_1_3_1_3_1"/><span class="lab"><font color="Green" title="E70">A168</font></span>: 
( <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E56">A137</font></span></a>, <a class="txt" href="#E3:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E62">A160</font></span></a>, <a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E49">A92</font></span></a>, <a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/>
<a NAME="E6:48_1_4_1_2_1_3_1_3_1"/><span class="lab"><font color="Green" title="E71">A169</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c17">y2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T29" target="_self" onmouseover="rs('surreali/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E7:48_1_4_1_2_1_3_1_3_1"/><span class="lab"><font color="Green" title="E72">A170</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c12">y1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T29" target="_self" onmouseover="rs('surreali/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E8:48_1_4_1_2_1_3_1_3_1"/><span class="lab"><font color="Green" title="E73">A171</font></span>: 
( <font color="Maroon" title="c19">x2</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp; <font color="Maroon" title="c14">x1</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E63">A161</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E57">A138</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E9:48_1_4_1_2_1_3_1_3_1"/><span class="lab"><font color="Green" title="E74">A172</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_4_1_2_1_3_1_3_1"><span class="lab"><font color="Green" title="E69">A167</font></span></a>, <a class="txt" href="#E5:48_1_4_1_2_1_3_1_3_1"><span class="lab"><font color="Green" title="E70">A168</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/>
<a NAME="E10:48_1_4_1_2_1_3_1_3_1"/><span class="lab"><font color="Green" title="E75">A173</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_4_1_2_1_3_1_3_1"><span class="lab"><font color="Green" title="E69">A167</font></span></a>, <a class="txt" href="#E5:48_1_4_1_2_1_3_1_3_1"><span class="lab"><font color="Green" title="E70">A168</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_3_1_3_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_4_1_2_1_3_1_3_1_1"/>
( <font color="Maroon" title="c12">y1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c17">y2</font> or <font color="Maroon" title="c12">y1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c17">y2</font> or <font color="Maroon" title="c17">y2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">y1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_3_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_3_1_3_1_1_1"/>
<font color="Maroon" title="c12">y1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c17">y2</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="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_3_1_3_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c17">y2</font> &amp; <font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c17">y2</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T43" onmouseover="rs('surrealr/T43')" onmouseout="rh()">SURREALR:43</a>, <a class="ref" href="surrealr.html#T39" onmouseover="rs('surrealr/T39')" onmouseout="rh()">SURREALR:39</a></span>;<br/><a NAME="E3:48_1_4_1_2_1_3_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><a NAME="E4:48_1_4_1_2_1_3_1_3_1_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font> &amp; <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a></span>;<br/><a NAME="E5:48_1_4_1_2_1_3_1_3_1_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> &amp; <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a></span>;<br/><a NAME="E6:48_1_4_1_2_1_3_1_3_1_1_1"/><span class="kw">then </span>
<a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_4_1_2_1_3_1_3_1"><span class="lab"><font color="Green" title="E74">A172</font></span></a>, <a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_1_4_1_2_1_3_1_3_1_1_1"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3_1_3_1"><span class="lab"><font color="Green" title="E71">A169</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_3_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_3_1_3_1_1_2"/>
<font color="Maroon" title="c12">y1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c17">y2</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="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_3_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c17">y2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/><a NAME="E3:48_1_4_1_2_1_3_1_3_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_3_1_3_1"><span class="lab"><font color="Green" title="E73">A171</font></span></a>, <a class="ref" href="surrealr.html#T74" onmouseover="rs('surrealr/T74')" onmouseout="rh()">SURREALR:74</a></span>;<br/><a NAME="E4:48_1_4_1_2_1_3_1_3_1_1_2"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_4_1_2_1_3_1_3_1"><span class="lab"><font color="Green" title="E68">A166</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E5:48_1_4_1_2_1_3_1_3_1_1_2"/><span class="kw">then </span>
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> &amp; <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">y1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_4_1_2_1_3_1_3_1"><span class="lab"><font color="Green" title="E74">A172</font></span></a>, <a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_1_4_1_2_1_3_1_3_1_1_2"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3_1_3_1"><span class="lab"><font color="Green" title="E71">A169</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2_1_3_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:48_1_4_1_2_1_3_1_3_1_1_3"/>
<font color="Maroon" title="c17">y2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">y1</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="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span></span><br/><div class="add"><a NAME="E2:48_1_4_1_2_1_3_1_3_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c12">y1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/><a NAME="E3:48_1_4_1_2_1_3_1_3_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_3_1_3_1"><span class="lab"><font color="Green" title="E73">A171</font></span></a>, <a class="ref" href="surrealr.html#T74" onmouseover="rs('surrealr/T74')" onmouseout="rh()">SURREALR:74</a></span>;<br/><a NAME="E4:48_1_4_1_2_1_3_1_3_1_1_3"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_4_1_2_1_3_1_3_1"><span class="lab"><font color="Green" title="E68">A166</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E5:48_1_4_1_2_1_3_1_3_1_1_3"/><span class="kw">then </span>
<a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c19">x2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">y2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1_4_1_2_1_3_1_3_1"><span class="lab"><font color="Green" title="E75">A173</font></span></a>, <a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_1_4_1_2_1_3_1_3_1_1_3"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_4_1_2_1_3_1_3_1"><span class="lab"><font color="Green" title="E72">A170</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E16:48_1_4_1_2_1_3_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c19">x2</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c14">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a></span>;<br/><a NAME="E17:48_1_4_1_2_1_3_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c12">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c17">y2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c20">Ix2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E64">A162</font></span></a>, <a class="txt" href="#E8:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E58">A139</font></span></a>, <a class="txt" href="#E12:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E60">A141</font></span></a>, <a class="txt" href="#E12:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E66">A164</font></span></a>, <a class="ref" href="surreali.html#T28" target="_self" onmouseover="rs('surreali/T28')" onmouseout="rh()">Th28</a></span>;<br/><span class="kw">hence </span><a NAME="E18:48_1_4_1_2_1_3_1_3"/>
 not <font color="Maroon" title="c6">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c5">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E57">A138</font></span></a>, <a class="txt" href="#E6:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E63">A161</font></span></a>, <a class="txt" href="#E15:48_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E61">A142</font></span></a>, <a class="txt" href="#E14:48_1_4_1_2_1_3_1_3"><span class="lab"><font color="Green" title="E67">A165</font></span></a>, <a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c5">UL</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>, <font color="Maroon" title="c6">UR</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> as   <a href="surreal0.html#V3" title="SURREAL0:attr.3">surreal-membered</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="surreali.html#T10" target="_self" onmouseover="rs('surreali/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">M</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E13:48_1"/><span class="lab"><font color="Green" title="E52">A174</font></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> <font color="Maroon" title="c5">UL</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">UR</font> holds <br/> ex <font color="Olive" title="b2">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Olive" title="b2">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#T47" onmouseover="rs('surreal0/T47')" onmouseout="rh()">SURREAL0:47</a></span>;<br/>
<a NAME="E14:48_1"/><span class="lab"><font color="Green" title="E53">A175</font></span>: 
 <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c2">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"><font color="Maroon" title="c5">UL</font>,<font color="Maroon" title="c6">UR</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surreali.html#T26" target="_self" onmouseover="rs('surreali/T26')" onmouseout="rh()">Th26</a></span>;<br/>
<a NAME="E15:48_1"/>
 <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <font color="Maroon" title="c7">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:48_1"><span class="lab"><font color="Green" title="E53">A175</font></span></a>, <a class="txt" href="#E13:48_1"><span class="lab"><font color="Green" title="E52">A174</font></span></a>, <a class="txt" href="#E10:48_1"><span class="lab"><font color="Green" title="E51">A94</font></span></a>, <a class="ref" href="surreal0.html#T46" onmouseover="rs('surreal0/T46')" onmouseout="rh()">SURREAL0:46</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:48_1"/>
 <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c2">x</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">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">  for <font color="Olive" title="b1">y</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c2">x</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b1">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">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="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c2">x</font> implies <font color="Maroon" title="c2">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E17:48_1"/><span class="lab"><font color="Green" title="E54">A176</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c2">x</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="c2">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">xy</font> = <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font>;<br/>
<a NAME="E18:48_1"/><span class="lab"><font color="Green" title="E55">A177</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</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">(<span class="default"><span class="p3">(<span class="default"><a href="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c8">y</font></span>)</span>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c8">y</font></span>)</span>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c8">y</font></span>)</span>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c8">y</font></span>)</span>)</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T50" onmouseover="rs('surrealr/T50')" onmouseout="rh()">SURREALR:50</a></span>;<br/>
<a NAME="E19:48_1"/><span class="lab"><font color="Green" title="E56">A178</font></span>: 
( <font color="Maroon" title="c5">UL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c8">y</font> &amp;  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c8">y</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="c8">y</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="c8">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>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c8">y</font> &amp;  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">UR</font> &amp; <font color="Maroon" title="c8">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">y</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>, <a class="txt" href="#E14:48_1"><span class="lab"><font color="Green" title="E53">A175</font></span></a>, <a class="txt" href="#E17:48_1"><span class="lab"><font color="Green" title="E54">A176</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E20:48_1"/><span class="lab"><font color="Green" title="E57">A179</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> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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>, <a class="ref" href="surreali.html#D6" target="_self" onmouseover="rs('surreali/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E21:48_1"/><span class="lab"><font color="Green" title="E58">A180</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a></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> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">r</font>, <font color="Maroon" title="c11">l</font> be   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="surreal0.html#D20" onmouseover="rs('surreal0/D20')" onmouseout="rh()">SURREAL0:def 20</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"> (  not <font color="Maroon" title="c10">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c11">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span> or  not <font color="Maroon" title="c11">l</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1_5"/><span class="lab"><font color="Green" title="E59">A181</font></span>: 
( <font color="Maroon" title="c10">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</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">  not <font color="Maroon" title="c11">l</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">r</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_5_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_5_1"/>
( <font color="Maroon" title="c11">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c8">y</font></span>)</span>) or <font color="Maroon" title="c11">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c8">y</font></span>)</span>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48_1"><span class="lab"><font color="Green" title="E55">A177</font></span></a>, <a class="txt" href="#E1:48_1_5"><span class="lab"><font color="Green" title="E59">A181</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_5_1_1"/>
<font color="Maroon" title="c11">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c8">y</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">  not <font color="Maroon" title="c11">l</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">r</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c13">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a><span class="kw"> such that </span><br/><a NAME="E3:48_1_5_1_1"/><span class="lab"><font color="Green" title="E60">A182</font></span>: 
( <font color="Maroon" title="c11">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p1">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <font color="Maroon" title="c13">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c8">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#D14" onmouseover="rs('surrealr/D14')" onmouseout="rh()">SURREALR:def 14</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:48_1_5_1_1"/><span class="lab"><font color="Green" title="E61">A183</font></span>: 
( <font color="Maroon" title="c13">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> &amp; (  for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c13">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_5_1_1"><span class="lab"><font color="Green" title="E60">A182</font></span></a>, <a class="txt" href="#E14:48_1"><span class="lab"><font color="Green" title="E53">A175</font></span></a>, <a class="txt" href="#E17:48_1"><span class="lab"><font color="Green" title="E54">A176</font></span></a>, <a class="ref" href="surreali.html#T27" target="_self" onmouseover="rs('surreali/T27')" onmouseout="rh()">Th27</a>, <a class="txt" href="#E20:48_1"><span class="lab"><font color="Green" title="E57">A179</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_5_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_5_1_1_1"/>
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  or <font color="Maroon" title="c12">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_5_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_5_1_1_1_1"/>
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</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">  not <font color="Maroon" title="c11">l</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">r</font></span><br/><div class="add"><a NAME="E2:48_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A184</font></span>: 
<font color="Maroon" title="c11">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T23" onmouseover="rs('surrealr/T23')" onmouseout="rh()">SURREALR:23</a>, <a class="txt" href="#E3:48_1_5_1_1"><span class="lab"><font color="Green" title="E60">A182</font></span></a></span>;<br/><a NAME="E3:48_1_5_1_1_1_1"/>
 <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_5_1_1"><span class="lab"><font color="Green" title="E61">A183</font></span></a>, <a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E49">A92</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_1_5_1_1_1_1"/>
 not <font color="Maroon" title="c11">l</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E62">A184</font></span></a>, <a class="txt" href="#E1:48_1_5"><span class="lab"><font color="Green" title="E59">A181</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_5_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_5_1_1_1_2"/><span class="lab"><font color="Green" title="E62">A185</font></span>: 
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</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">  not <font color="Maroon" title="c11">l</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">r</font></span><br/><div class="add"><a NAME="E2:48_1_5_1_1_1_2"/>
<font color="Maroon" title="c12">x1</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_5_1_1"><span class="lab"><font color="Green" title="E60">A182</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E3:48_1_5_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A186</font></span>: 
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c12">x1</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a>, <a class="ref" href="surreali.html#T22" target="_self" onmouseover="rs('surreali/T22')" onmouseout="rh()">Th22</a></span>;<br/><a NAME="E4:48_1_5_1_1_1_2"/><span class="lab"><font color="Green" title="E64">A187</font></span>: 
<font color="Maroon" title="c12">x1</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E62">A185</font></span></a>, <a class="txt" href="#E3:48_1_5_1_1"><span class="lab"><font color="Green" title="E60">A182</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">Ix1</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c12">x1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E3:48_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E63">A186</font></span></a></span>;<br/><a NAME="E6:48_1_5_1_1_1_2"/>
<font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E3:48_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E63">A186</font></span></a>, <a class="txt" href="#E4:48_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E64">A187</font></span></a></span>;<br/><a NAME="E7:48_1_5_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A188</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T30" target="_self" onmouseover="rs('surreali/T30')" onmouseout="rh()">Th30</a></span>;<br/><a NAME="E8:48_1_5_1_1_1_2"/>
<font color="Maroon" title="c12">x1</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_5_1_1"><span class="lab"><font color="Green" title="E60">A182</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E9:48_1_5_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="txt" href="#E1:48_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E62">A185</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E10:48_1_5_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">Ix1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E11:48_1_5_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c13">y1</font>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E62">A185</font></span></a>, <a class="txt" href="#E3:48_1_5_1_1"><span class="lab"><font color="Green" title="E60">A182</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/><a NAME="E12:48_1_5_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_5_1_1"><span class="lab"><font color="Green" title="E61">A183</font></span></a>, <a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/><a NAME="E13:48_1_5_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></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"><span class="p4">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p5">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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"><span class="p3">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E14:48_1_5_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A189</font></span>: 
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T6" target="_self" onmouseover="rs('surreali/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E15:48_1_5_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">UL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1"><span class="lab"><font color="Green" title="E57">A179</font></span></a>, <a class="txt" href="#E14:48_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E66">A189</font></span></a>, <a class="ref" href="card_5.html#T2" onmouseover="rs('card_5/T2')" onmouseout="rh()">CARD_5:2</a></span>;<br/><a NAME="E16:48_1_5_1_1_1_2"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c8">y</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:48_1"><span class="lab"><font color="Green" title="E56">A178</font></span></a>, <a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a></span>;<br/><a NAME="E17:48_1_5_1_1_1_2"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E64">A187</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E18:48_1_5_1_1_1_2"/><span class="kw">then </span>
<a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><a NAME="E19:48_1_5_1_1_1_2"/><span class="kw">then </span>
(  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> &amp; <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c11">l</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_5_1_1"><span class="lab"><font color="Green" title="E60">A182</font></span></a>, <a class="txt" href="#E7:48_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E65">A188</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><span class="kw">hence </span><a NAME="E20:48_1_5_1_1_1_2"/>
 not <font color="Maroon" title="c11">l</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_5"><span class="lab"><font color="Green" title="E59">A181</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_5_1_2"/>
<font color="Maroon" title="c11">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c8">y</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">  not <font color="Maroon" title="c11">l</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">r</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c13">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a><span class="kw"> such that </span><br/><a NAME="E3:48_1_5_1_2"/><span class="lab"><font color="Green" title="E60">A190</font></span>: 
( <font color="Maroon" title="c11">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p1">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <font color="Maroon" title="c13">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c8">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#D14" onmouseover="rs('surrealr/D14')" onmouseout="rh()">SURREALR:def 14</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:48_1_5_1_2"/><span class="lab"><font color="Green" title="E61">A191</font></span>: 
( <font color="Maroon" title="c13">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> &amp; (  for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c13">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_5_1_2"><span class="lab"><font color="Green" title="E60">A190</font></span></a>, <a class="txt" href="#E14:48_1"><span class="lab"><font color="Green" title="E53">A175</font></span></a>, <a class="txt" href="#E17:48_1"><span class="lab"><font color="Green" title="E54">A176</font></span></a>, <a class="ref" href="surreali.html#T27" target="_self" onmouseover="rs('surreali/T27')" onmouseout="rh()">Th27</a>, <a class="txt" href="#E20:48_1"><span class="lab"><font color="Green" title="E57">A179</font></span></a></span>;<br/><a NAME="E6:48_1_5_1_2"/>
<font color="Maroon" title="c12">x1</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_5_1_2"><span class="lab"><font color="Green" title="E60">A190</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E7:48_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A192</font></span>: 
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c12">x1</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a>, <a class="ref" href="surreali.html#T22" target="_self" onmouseover="rs('surreali/T22')" onmouseout="rh()">Th22</a></span>;<br/><a NAME="E8:48_1_5_1_2"/><span class="lab"><font color="Green" title="E63">A193</font></span>: 
<font color="Maroon" title="c12">x1</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E3:48_1_5_1_2"><span class="lab"><font color="Green" title="E60">A190</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">Ix1</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c12">x1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E7:48_1_5_1_2"><span class="lab"><font color="Green" title="E62">A192</font></span></a></span>;<br/><a NAME="E10:48_1_5_1_2"/>
<font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E7:48_1_5_1_2"><span class="lab"><font color="Green" title="E62">A192</font></span></a>, <a class="txt" href="#E8:48_1_5_1_2"><span class="lab"><font color="Green" title="E63">A193</font></span></a></span>;<br/><a NAME="E11:48_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A194</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T30" target="_self" onmouseover="rs('surreali/T30')" onmouseout="rh()">Th30</a></span>;<br/><a NAME="E12:48_1_5_1_2"/><span class="lab"><font color="Green" title="E65">A195</font></span>: 
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp;  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">x1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_5_1_2"><span class="lab"><font color="Green" title="E63">A193</font></span></a></span>;<br/><a NAME="E13:48_1_5_1_2"/>
<font color="Maroon" title="c12">x1</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_5_1_2"><span class="lab"><font color="Green" title="E60">A190</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E14:48_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="txt" href="#E12:48_1_5_1_2"><span class="lab"><font color="Green" title="E65">A195</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E15:48_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">Ix1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E16:48_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c13">y1</font>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48_1_5_1_2"><span class="lab"><font color="Green" title="E65">A195</font></span></a>, <a class="txt" href="#E3:48_1_5_1_2"><span class="lab"><font color="Green" title="E60">A190</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/><a NAME="E17:48_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_5_1_2"><span class="lab"><font color="Green" title="E61">A191</font></span></a>, <a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/><a NAME="E18:48_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></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"><span class="p3">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E19:48_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></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"><span class="p4">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p5">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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"><span class="p3">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E20:48_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A196</font></span>: 
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T6" target="_self" onmouseover="rs('surreali/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E21:48_1_5_1_2"/>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">UL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1"><span class="lab"><font color="Green" title="E57">A179</font></span></a>, <a class="txt" href="#E20:48_1_5_1_2"><span class="lab"><font color="Green" title="E66">A196</font></span></a>, <a class="ref" href="card_5.html#T2" onmouseover="rs('card_5/T2')" onmouseout="rh()">CARD_5:2</a></span>;<br/><a NAME="E22:48_1_5_1_2"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c8">y</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:48_1"><span class="lab"><font color="Green" title="E56">A178</font></span></a>, <a class="ref" href="surrealr.html#T45" onmouseover="rs('surrealr/T45')" onmouseout="rh()">SURREALR:45</a></span>;<br/><a NAME="E23:48_1_5_1_2"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_5_1_2"><span class="lab"><font color="Green" title="E63">A193</font></span></a>, <a class="ref" href="surrealr.html#T72" onmouseover="rs('surrealr/T72')" onmouseout="rh()">SURREALR:72</a></span>;<br/><a NAME="E24:48_1_5_1_2"/><span class="kw">then </span>
<a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><a NAME="E25:48_1_5_1_2"/><span class="kw">then </span>
(  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> &amp; <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c11">l</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_5_1_2"><span class="lab"><font color="Green" title="E64">A194</font></span></a>, <a class="txt" href="#E3:48_1_5_1_2"><span class="lab"><font color="Green" title="E60">A190</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><span class="kw">hence </span><a NAME="E26:48_1_5_1_2"/>
 not <font color="Maroon" title="c11">l</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_5"><span class="lab"><font color="Green" title="E59">A181</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E22:48_1"/><span class="lab"><font color="Green" title="E59">A197</font></span>: 
 <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</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="surrealo.html#K1" title="SURREALO:func.1">1_No</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">l</font>, <font color="Maroon" title="c11">r</font> be   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="surreal0.html#D20" onmouseover="rs('surreal0/D20')" onmouseout="rh()">SURREAL0:def 20</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"> (  not <font color="Maroon" title="c10">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span> or  not <font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c11">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1_6"/><span class="lab"><font color="Green" title="E60">A198</font></span>: 
( <font color="Maroon" title="c10">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span> &amp; <font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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">  not <font color="Maroon" title="c11">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">l</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_6_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_6_1"/>
( <font color="Maroon" title="c10">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c8">y</font></span>)</span>) or <font color="Maroon" title="c10">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c8">y</font></span>)</span>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48_1"><span class="lab"><font color="Green" title="E55">A177</font></span></a>, <a class="txt" href="#E1:48_1_6"><span class="lab"><font color="Green" title="E60">A198</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_6_1_1"/>
<font color="Maroon" title="c10">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c8">y</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">  not <font color="Maroon" title="c11">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">l</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c13">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a><span class="kw"> such that </span><br/><a NAME="E3:48_1_6_1_1"/><span class="lab"><font color="Green" title="E61">A199</font></span>: 
( <font color="Maroon" title="c10">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p1">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <font color="Maroon" title="c13">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c8">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#D14" onmouseover="rs('surrealr/D14')" onmouseout="rh()">SURREALR:def 14</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:48_1_6_1_1"/><span class="lab"><font color="Green" title="E62">A200</font></span>: 
( <font color="Maroon" title="c13">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> &amp; (  for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c13">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_6_1_1"><span class="lab"><font color="Green" title="E61">A199</font></span></a>, <a class="txt" href="#E14:48_1"><span class="lab"><font color="Green" title="E53">A175</font></span></a>, <a class="txt" href="#E17:48_1"><span class="lab"><font color="Green" title="E54">A176</font></span></a>, <a class="ref" href="surreali.html#T27" target="_self" onmouseover="rs('surreali/T27')" onmouseout="rh()">Th27</a>, <a class="txt" href="#E20:48_1"><span class="lab"><font color="Green" title="E57">A179</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_6_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_6_1_1_1"/>
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  or <font color="Maroon" title="c12">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_6_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_6_1_1_1_1"/>
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</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">  not <font color="Maroon" title="c11">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">l</font></span><br/><div class="add"><a NAME="E2:48_1_6_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A201</font></span>: 
<font color="Maroon" title="c10">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T23" onmouseover="rs('surrealr/T23')" onmouseout="rh()">SURREALR:23</a>, <a class="txt" href="#E3:48_1_6_1_1"><span class="lab"><font color="Green" title="E61">A199</font></span></a></span>;<br/><a NAME="E3:48_1_6_1_1_1_1"/>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E49">A92</font></span></a>, <a class="txt" href="#E5:48_1_6_1_1"><span class="lab"><font color="Green" title="E62">A200</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_1_6_1_1_1_1"/>
 not <font color="Maroon" title="c11">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E63">A201</font></span></a>, <a class="txt" href="#E1:48_1_6"><span class="lab"><font color="Green" title="E60">A198</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_6_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_6_1_1_1_2"/><span class="lab"><font color="Green" title="E63">A202</font></span>: 
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</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">  not <font color="Maroon" title="c11">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">l</font></span><br/><div class="add"><a NAME="E2:48_1_6_1_1_1_2"/>
<font color="Maroon" title="c12">x1</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_6_1_1"><span class="lab"><font color="Green" title="E61">A199</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E3:48_1_6_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A203</font></span>: 
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c12">x1</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a>, <a class="ref" href="surreali.html#T22" target="_self" onmouseover="rs('surreali/T22')" onmouseout="rh()">Th22</a></span>;<br/><a NAME="E4:48_1_6_1_1_1_2"/><span class="lab"><font color="Green" title="E65">A204</font></span>: 
<font color="Maroon" title="c12">x1</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E63">A202</font></span></a>, <a class="txt" href="#E3:48_1_6_1_1"><span class="lab"><font color="Green" title="E61">A199</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">Ix1</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c12">x1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E3:48_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E64">A203</font></span></a></span>;<br/><a NAME="E6:48_1_6_1_1_1_2"/>
<font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E3:48_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E64">A203</font></span></a>, <a class="txt" href="#E4:48_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E65">A204</font></span></a></span>;<br/><a NAME="E7:48_1_6_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A205</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T30" target="_self" onmouseover="rs('surreali/T30')" onmouseout="rh()">Th30</a></span>;<br/><a NAME="E8:48_1_6_1_1_1_2"/>
<font color="Maroon" title="c12">x1</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_6_1_1"><span class="lab"><font color="Green" title="E61">A199</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E9:48_1_6_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="txt" href="#E1:48_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E63">A202</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E10:48_1_6_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">Ix1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E11:48_1_6_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c13">y1</font>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E63">A202</font></span></a>, <a class="txt" href="#E3:48_1_6_1_1"><span class="lab"><font color="Green" title="E61">A199</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/><a NAME="E12:48_1_6_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_6_1_1"><span class="lab"><font color="Green" title="E62">A200</font></span></a>, <a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/><a NAME="E13:48_1_6_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></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"><span class="p3">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E14:48_1_6_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></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"><span class="p4">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p5">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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"><span class="p3">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E15:48_1_6_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A206</font></span>: 
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T6" target="_self" onmouseover="rs('surreali/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E16:48_1_6_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">UR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1"><span class="lab"><font color="Green" title="E57">A179</font></span></a>, <a class="txt" href="#E15:48_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E67">A206</font></span></a>, <a class="ref" href="card_5.html#T2" onmouseover="rs('card_5/T2')" onmouseout="rh()">CARD_5:2</a></span>;<br/><a NAME="E17:48_1_6_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:48_1"><span class="lab"><font color="Green" title="E56">A178</font></span></a>, <a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/><a NAME="E18:48_1_6_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E65">A204</font></span></a>, <a class="ref" href="surrealr.html#T74" onmouseover="rs('surrealr/T74')" onmouseout="rh()">SURREALR:74</a></span>;<br/><a NAME="E19:48_1_6_1_1_1_2"/><span class="kw">then </span>
<a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><a NAME="E20:48_1_6_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c10">l</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> &amp; <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</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="txt" href="#E3:48_1_6_1_1"><span class="lab"><font color="Green" title="E61">A199</font></span></a>, <a class="txt" href="#E7:48_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E66">A205</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><span class="kw">hence </span><a NAME="E21:48_1_6_1_1_1_2"/>
 not <font color="Maroon" title="c11">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_6"><span class="lab"><font color="Green" title="E60">A198</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_6_1_2"/>
<font color="Maroon" title="c10">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c8">y</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">  not <font color="Maroon" title="c11">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">l</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c13">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a><span class="kw"> such that </span><br/><a NAME="E3:48_1_6_1_2"/><span class="lab"><font color="Green" title="E61">A207</font></span>: 
( <font color="Maroon" title="c10">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p1">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp; <font color="Maroon" title="c13">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c8">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#D14" onmouseover="rs('surrealr/D14')" onmouseout="rh()">SURREALR:def 14</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:48_1_6_1_2"/><span class="lab"><font color="Green" title="E62">A208</font></span>: 
( <font color="Maroon" title="c13">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> &amp; (  for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c13">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_6_1_2"><span class="lab"><font color="Green" title="E61">A207</font></span></a>, <a class="txt" href="#E14:48_1"><span class="lab"><font color="Green" title="E53">A175</font></span></a>, <a class="txt" href="#E17:48_1"><span class="lab"><font color="Green" title="E54">A176</font></span></a>, <a class="ref" href="surreali.html#T27" target="_self" onmouseover="rs('surreali/T27')" onmouseout="rh()">Th27</a>, <a class="txt" href="#E20:48_1"><span class="lab"><font color="Green" title="E57">A179</font></span></a></span>;<br/><a NAME="E6:48_1_6_1_2"/>
<font color="Maroon" title="c12">x1</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_6_1_2"><span class="lab"><font color="Green" title="E61">A207</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E7:48_1_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A209</font></span>: 
(  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c12">x1</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> &amp;  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a>, <a class="ref" href="surreali.html#T22" target="_self" onmouseover="rs('surreali/T22')" onmouseout="rh()">Th22</a></span>;<br/><a NAME="E8:48_1_6_1_2"/><span class="lab"><font color="Green" title="E64">A210</font></span>: 
<font color="Maroon" title="c12">x1</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E3:48_1_6_1_2"><span class="lab"><font color="Green" title="E61">A207</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">Ix1</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c12">x1</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E7:48_1_6_1_2"><span class="lab"><font color="Green" title="E63">A209</font></span></a></span>;<br/><a NAME="E10:48_1_6_1_2"/>
<font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E7:48_1_6_1_2"><span class="lab"><font color="Green" title="E63">A209</font></span></a>, <a class="txt" href="#E8:48_1_6_1_2"><span class="lab"><font color="Green" title="E64">A210</font></span></a></span>;<br/><a NAME="E11:48_1_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A211</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T30" target="_self" onmouseover="rs('surreali/T30')" onmouseout="rh()">Th30</a></span>;<br/><a NAME="E12:48_1_6_1_2"/><span class="lab"><font color="Green" title="E66">A212</font></span>: 
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#R1" title="SURREALO:pred.1">&lt;=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp;  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c12">x1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_6_1_2"><span class="lab"><font color="Green" title="E64">A210</font></span></a></span>;<br/><a NAME="E13:48_1_6_1_2"/>
<font color="Maroon" title="c12">x1</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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_6_1_2"><span class="lab"><font color="Green" title="E61">A207</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E14:48_1_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></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="txt" href="#E12:48_1_6_1_2"><span class="lab"><font color="Green" title="E66">A212</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E15:48_1_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">Ix1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E16:48_1_6_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c13">y1</font>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48_1_6_1_2"><span class="lab"><font color="Green" title="E66">A212</font></span></a>, <a class="txt" href="#E3:48_1_6_1_2"><span class="lab"><font color="Green" title="E61">A207</font></span></a>, <a class="ref" href="surreali.html#D2" target="_self" onmouseover="rs('surreali/D2')" onmouseout="rh()">Def2</a></span>;<br/><a NAME="E17:48_1_6_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_6_1_2"><span class="lab"><font color="Green" title="E62">A208</font></span></a>, <a class="ref" href="surreali.html#D3" target="_self" onmouseover="rs('surreali/D3')" onmouseout="rh()">Def3</a></span>;<br/><a NAME="E18:48_1_6_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></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"><span class="p4">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p5">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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"><span class="p3">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E19:48_1_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A213</font></span>: 
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T6" target="_self" onmouseover="rs('surreali/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E20:48_1_6_1_2"/>
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c15">Ix1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">UR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1"><span class="lab"><font color="Green" title="E57">A179</font></span></a>, <a class="txt" href="#E19:48_1_6_1_2"><span class="lab"><font color="Green" title="E67">A213</font></span></a>, <a class="ref" href="card_5.html#T2" onmouseover="rs('card_5/T2')" onmouseout="rh()">CARD_5:2</a></span>;<br/><a NAME="E21:48_1_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:48_1"><span class="lab"><font color="Green" title="E56">A178</font></span></a>, <a class="ref" href="surrealr.html#T46" onmouseover="rs('surrealr/T46')" onmouseout="rh()">SURREALR:46</a></span>;<br/><a NAME="E22:48_1_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_6_1_2"><span class="lab"><font color="Green" title="E64">A210</font></span></a>, <a class="ref" href="surrealr.html#T74" onmouseover="rs('surrealr/T74')" onmouseout="rh()">SURREALR:74</a></span>;<br/><a NAME="E23:48_1_6_1_2"/><span class="kw">then </span>
<a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c15">Ix1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T44" onmouseover="rs('surrealr/T44')" onmouseout="rh()">SURREALR:44</a></span>;<br/><a NAME="E24:48_1_6_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c10">l</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> &amp; <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</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="txt" href="#E3:48_1_6_1_2"><span class="lab"><font color="Green" title="E61">A207</font></span></a>, <a class="txt" href="#E11:48_1_6_1_2"><span class="lab"><font color="Green" title="E65">A211</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/><span class="kw">hence </span><a NAME="E25:48_1_6_1_2"/>
 not <font color="Maroon" title="c11">r</font> <a href="surreal0.html#R5" title="SURREAL0:pred.5">&lt;=</a> <font color="Maroon" title="c10">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_6"><span class="lab"><font color="Green" title="E60">A198</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E23:48_1"/><span class="lab"><font color="Green" title="E60">A214</font></span>: 
<span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a></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> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1">(<span class="default"><span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span>
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">oxy</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span>,<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"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_1"><span class="lab"><font color="Green" title="E58">A180</font></span></a>, <a class="txt" href="#E22:48_1"><span class="lab"><font color="Green" title="E59">A197</font></span></a>, <a class="txt" href="#E23:48_1"><span class="lab"><font color="Green" title="E60">A214</font></span></a>, <a class="ref" href="surrealo.html#T14" onmouseover="rs('surrealo/T14')" onmouseout="rh()">SURREALO:14</a></span>;<br/>
<a NAME="E25:48_1"/><span class="lab"><font color="Green" title="E61">A215</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surreali.html#D9" target="_self" onmouseover="rs('surreali/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E26:48_1"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E27:48_1"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="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> (<font color="Maroon" title="c5">UL</font>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E28:48_1"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><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="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> (<font color="Maroon" title="c5">UL</font>,<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</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> (<font color="Maroon" title="c6">UR</font>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E29:48_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A216</font></span>: 
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c8">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:48_1"><span class="lab"><font color="Green" title="E54">A176</font></span></a>, <a class="txt" href="#E14:48_1"><span class="lab"><font color="Green" title="E53">A175</font></span></a>, <a class="ref" href="surreali.html#T12" target="_self" onmouseover="rs('surreali/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<a NAME="E30:48_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a></span>)</span></span>)</span> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2">(<span class="default"><a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 
;<br/>
<a NAME="E31:48_1"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p1">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c8">y</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:48_1"><span class="lab"><font color="Green" title="E62">A216</font></span></a>, <a class="txt" href="#E25:48_1"><span class="lab"><font color="Green" title="E61">A215</font></span></a>, <a class="ref" href="surrealr.html#D14" onmouseover="rs('surrealr/D14')" onmouseout="rh()">SURREALR:def 14</a></span>;<br/>
<a NAME="E32:48_1"/><span class="kw">then </span>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p2">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c8">y</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="surrealr.html#K15" title="SURREALR:func.15">comp</a> (<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c8">y</font>,<span class="p2">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c8">y</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E33:48_1"/><span class="kw">then </span>
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">oxy</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48_1"><span class="lab"><font color="Green" title="E55">A177</font></span></a>, <a class="ref" href="zfmisc_1.html#T40" onmouseover="rs('zfmisc_1/T40')" onmouseout="rh()">ZFMISC_1:40</a></span>;<br/>
<a NAME="E34:48_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A217</font></span>: 
<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_1"><span class="lab"><font color="Green" title="E58">A180</font></span></a>, <a class="txt" href="#E22:48_1"><span class="lab"><font color="Green" title="E59">A197</font></span></a>, <a class="txt" href="#E23:48_1"><span class="lab"><font color="Green" title="E60">A214</font></span></a>, <a class="ref" href="surrealo.html#T15" onmouseover="rs('surrealo/T15')" onmouseout="rh()">SURREALO:15</a></span>;<br/>
<a NAME="E35:48_1"/>
<font color="Maroon" title="c2">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="surreali.html#T18" target="_self" onmouseover="rs('surreali/T18')" onmouseout="rh()">Th18</a>, <a class="ref" href="surrealr.html#T51" onmouseover="rs('surrealr/T51')" onmouseout="rh()">SURREALR:51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E36:48_1"/>
<font color="Maroon" title="c2">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c8">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:48_1"><span class="lab"><font color="Green" title="E63">A217</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="E2:48"/>
 for <font color="Olive" title="b1">D</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">D</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal1.html#S2" onmouseover="rs('ordinal1/S2')" onmouseout="rh()">ORDINAL1:sch 2</a>(<a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E41">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E3:48"/>
 for <font color="Olive" title="b1">x</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  holds <br/>(  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Olive" title="b1">x</font> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  &amp; (  for <font color="Olive" title="b2">y</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Olive" title="b1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b2">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</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"> verum</span><br/>


</div>
