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<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Inv</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon" title="c1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp;  <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="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Inv</font> holds <br/> <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Olive" title="b1">Inv</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Olive" title="b1">Inv</font>)</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Inv</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp;  <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="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Inv</font> implies  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c2">Inv</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c2">Inv</font>)</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">A</font> =  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">Nx</font> = <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>;<br/>
<span class="kw">assume </span><a NAME="E1:43"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
( <font color="Maroon" title="c1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  &amp;  <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="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Inv</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c2">Inv</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c2">Inv</font>)</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">S</font> being   <a href="cohsp_1.html#V6" title="COHSP_1:attr.6">c=-monotone</a>   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="ordinal1.html#NM4" title="ORDINAL1:NM.4">Sequence</a><span class="kw"> such that </span><br/><a NAME="E3:43"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp;  <a href="surreali.html#K9" title="SURREALI:func.9">No_inverse_op</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E4:43"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
 for <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">SB</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="surreali.html#K8" title="SURREALI:func.8">Positives</a> <font color="Olive" title="b1">B</font> st <br/>( <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">SB</font> &amp; (  for <font color="Olive" title="b3">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K8" title="SURREALI:func.8">Positives</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Olive" title="b2">SB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">o</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"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="Olive" title="b3">o</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p5">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Olive" title="b1">B</font></span>)</span></span>)</span></span>)</span>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="Olive" title="b3">o</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p5">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Olive" title="b1">B</font></span>)</span></span>)</span></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="surreali.html#D11" target="_self" onmouseover="rs('surreali/D11')" onmouseout="rh()">Def11</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">SB</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="surreali.html#K8" title="SURREALI:func.8">Positives</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:43"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">SB</font>
 <span class="kw">and </span><br/><a NAME="E7:43"/><span class="lab"><font color="Green" title="E40">A5</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>  <a href="surreali.html#K8" title="SURREALI:func.8">Positives</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span> holds <br/><font color="Maroon" title="c6">SB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="Olive" title="b1">o</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p5">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="Olive" title="b1">o</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p5">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></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="txt" href="#E4:43"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="ordinal1.html#T6" onmouseover="rs('ordinal1/T6')" onmouseout="rh()">ORDINAL1:6</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">UA</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E8:43"/>
<font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#D18" onmouseover="rs('surreal0/D18')" onmouseout="rh()">SURREAL0:def 18</a></span>;<br/>
<a NAME="E9:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A6</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K8" title="SURREALI:func.8">Positives</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="surreali.html#D10" target="_self" onmouseover="rs('surreali/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">XX</font> = <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="c1">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="c1">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>;<br/>
<a NAME="E10:43"/><span class="lab"><font color="Green" title="E42">A7</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="c1">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="c1">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>  <a href="surreali.html#K8" title="SURREALI:func.8">Positives</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="surreali.html#T24" target="_self" onmouseover="rs('surreali/T24')" onmouseout="rh()">Th24</a></span>;<br/>
<a NAME="E11:43"/><span class="lab"><font color="Green" title="E43">A8</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="c1">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="c1">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="c1">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="surreali.html#T20" target="_self" onmouseover="rs('surreali/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E12:43"/>
<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="c1">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="c1">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>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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="c9">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="c1">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="c1">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> or <font color="Maroon" title="c9">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E44">A9</font></span>: 
<font color="Maroon" title="c9">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="c1">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="c1">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"> <font color="Maroon" title="c9">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">o</font> = <font color="Maroon" title="c9">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/>
<span class="kw">set </span><font color="Maroon" title="c11">b</font> =  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font>;<br/>
<a NAME="E3:43_1"/><span class="lab"><font color="Green" title="E45">A10</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="c10">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="ref" href="surreali.html#D10" target="_self" onmouseover="rs('surreali/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E4:43_1"/>
<font color="Maroon" title="c10">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#D18" onmouseover="rs('surreal0/D18')" onmouseout="rh()">SURREAL0:def 18</a></span>;<br/>
<a NAME="E5:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A11</font></span>: 
<font color="Maroon" title="c10">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K8" title="SURREALI:func.8">Positives</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="ref" href="surreali.html#D10" target="_self" onmouseover="rs('surreali/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E6:43_1"/><span class="lab"><font color="Green" title="E47">A12</font></span>: 
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">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="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a></span>;<br/>
<a NAME="E7:43_1"/>
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T8" onmouseover="rs('ordinal1/T8')" onmouseout="rh()">ORDINAL1:8</a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a></span>;<br/>
<a NAME="E8:43_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">SB</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="surreali.html#K8" title="SURREALI:func.8">Positives</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span> st <br/>( <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">SB</font> &amp; (  for <font color="Olive" title="b2">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K8" title="SURREALI:func.8">Positives</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span> holds <br/><font color="Olive" title="b1">SB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">o</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"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="Olive" title="b2">o</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p5">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="Olive" title="b2">o</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p5">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span></span>)</span></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="txt" href="#E4:43"><span class="lab"><font color="Green" title="E38">A3</font></span></a></span>;<br/>
<a NAME="E9:43_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:43_1"/>
<font color="Maroon" title="c9">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="ref" href="surrealr.html#T5" onmouseover="rs('surrealr/T5')" onmouseout="rh()">SURREALR:5</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="E13:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A13</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p5">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c1">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="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="c1">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="p3"><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>)</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"><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="c1">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="c1">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="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/>
<a NAME="E14:43"/>
( <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="c1">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="c1">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="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#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="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</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="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">Inv</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T11" onmouseover="rs('relat_1/T11')" onmouseout="rh()">RELAT_1:11</a>, <a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E15:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A14</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Inv</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c1">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="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="c1">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="p3"><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>)</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"><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="c1">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="c1">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="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/>
<a NAME="E16:43"/>
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">a</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="c1">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="c1">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> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p5">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c1">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="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="c1">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="p3"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Inv</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c1">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="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="c1">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="p3"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">a</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="c1">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="c1">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"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p5">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c1">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="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="c1">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="p3"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Inv</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c1">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="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="c1">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="p3"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_2"/><span class="lab"><font color="Green" title="E46">A15</font></span>: 
<font color="Maroon" title="c9">a</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="c1">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="c1">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"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p5">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c1">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="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="c1">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="p3"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Inv</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c1">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="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="c1">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="p3"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">o</font> = <font color="Maroon" title="c9">a</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="E3:43_2"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p5">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c1">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="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="c1">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="p3"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">b</font> =  <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font>;<br/>
<a NAME="E4:43_2"/><span class="lab"><font color="Green" title="E48">A17</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="c10">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="surreali.html#D10" target="_self" onmouseover="rs('surreali/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E5:43_2"/>
<font color="Maroon" title="c10">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K10" title="SURREAL0:func.10">Day</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreal0.html#D18" onmouseover="rs('surreal0/D18')" onmouseout="rh()">SURREAL0:def 18</a></span>;<br/>
<a NAME="E6:43_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A18</font></span>: 
<font color="Maroon" title="c10">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreali.html#K8" title="SURREALI:func.8">Positives</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="ref" href="surreali.html#D10" target="_self" onmouseover="rs('surreali/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E7:43_2"/><span class="lab"><font color="Green" title="E50">A19</font></span>: 
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">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="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a></span>;<br/>
<a NAME="E8:43_2"/><span class="lab"><font color="Green" title="E51">A20</font></span>: 
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T8" onmouseover="rs('ordinal1/T8')" onmouseout="rh()">ORDINAL1:8</a>, <a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a></span>;<br/>
<a NAME="E9:43_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">SB</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="surreali.html#K8" title="SURREALI:func.8">Positives</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span> st <br/>( <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">SB</font> &amp; (  for <font color="Olive" title="b2">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K8" title="SURREALI:func.8">Positives</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span> holds <br/><font color="Olive" title="b1">SB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">o</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"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="Olive" title="b2">o</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p5">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="Olive" title="b2">o</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p5">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span></span>)</span></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="txt" href="#E4:43"><span class="lab"><font color="Green" title="E38">A3</font></span></a></span>;<br/>
<a NAME="E10:43_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A21</font></span>: 
<font color="Maroon" title="c10">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E11:43_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A22</font></span>: 
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="ref" href="surrealr.html#T5" onmouseover="rs('surrealr/T5')" onmouseout="rh()">SURREALR:5</a></span>;<br/>
<a NAME="E12:43_2"/><span class="lab"><font color="Green" title="E54">A23</font></span>: 
 <a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T8" onmouseover="rs('ordinal1/T8')" onmouseout="rh()">ORDINAL1:8</a>, <a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="ref" href="surrealo.html#T1" onmouseover="rs('surrealo/T1')" onmouseout="rh()">SURREALO:1</a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E37">A2</font></span></a></span>;<br/>
<a NAME="E13:43_2"/><span class="lab"><font color="Green" title="E55">A24</font></span>: 
 <a href="surreali.html#K9" title="SURREALI:func.9">No_inverse_op</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c10">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="surreali.html#T25" target="_self" onmouseover="rs('surreali/T25')" onmouseout="rh()">Th25</a>, <a class="txt" href="#E8:43_2"><span class="lab"><font color="Green" title="E51">A20</font></span></a></span>;<br/>
<a NAME="E14:43_2"/><span class="lab"><font color="Green" title="E56">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">Inv</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c1">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="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="c1">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="p3"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Inv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E15:43_2"/><span class="lab"><font color="Green" title="E57">A26</font></span>: 
<font color="Maroon" title="c10">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="surreali.html#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="c1">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="#E1:43_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E16:43_2"/>
<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="c1">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">o</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="c10">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="surreali.html#D13" target="_self" onmouseover="rs('surreali/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E17:43_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c10">o</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c1">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="#E15:43_2"><span class="lab"><font color="Green" title="E57">A26</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E18:43_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">Inv</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c1">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="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="c1">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="p3"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</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="c10">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E14:43_2"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:43_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p5">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c1">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="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="c1">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="p3"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Inv</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c1">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="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="c1">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="p3"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_2"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E11:43_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E10:43_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E12:43_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="surrealr.html#T2" onmouseover="rs('surrealr/T2')" onmouseout="rh()">SURREALR:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:43"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<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> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">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="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="c1">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="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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Inv</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<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> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">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="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="c1">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="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></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E15:43"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>;<br/>
<a NAME="E18:43"/><span class="kw">then </span>
(  <a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c2">Inv</font>) &amp;  <a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><a href="surreal0.html#K12" title="SURREAL0:func.12">born</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c2">Inv</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T17" target="_self" onmouseover="rs('surreali/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:43"/>
 <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c2">Inv</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <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="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<font color="Maroon" title="c2">Inv</font>)</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="txt" href="#E9:43"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E37">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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