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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"> ( <font color="Maroon" title="c1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a>  implies <font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p5">(<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="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 class="p5">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<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 class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<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 class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<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="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 class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</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="c2">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">set </span><font color="Maroon" title="c3">Inv</font> =  <a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p1"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>;<br/>
<span class="kw">assume </span><a NAME="E1:56"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<font color="Maroon" title="c1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p5">(<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="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 class="p5">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<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 class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<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 class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<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="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 class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<a NAME="E2:56"/><span class="kw">then </span>
 not <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a> 
 
;<br/>
<a NAME="E3:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <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="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="surreali.html#D14" target="_self" onmouseover="rs('surreali/D14')" onmouseout="rh()">Def14</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">UL</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">UR</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>;<br/>
<a NAME="E4:56"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
 <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>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></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="#E1:56"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="surreali.html#T26" target="_self" onmouseover="rs('surreali/T26')" onmouseout="rh()">Th26</a></span>;<br/>
<a NAME="E5:56"/><span class="kw">then </span>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E6:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A4</font></span>: 
(  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span></span>)</span>,<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="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</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="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="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span></span>)</span>) &amp;  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span></span>)</span>,<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="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</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>,<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="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="surreali.html#T34" target="_self" onmouseover="rs('surreali/T34')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E7:56"/>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E8:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A5</font></span>: 
(  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span></span>)</span>,<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="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</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>,<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="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span></span>)</span>) &amp;  <a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span></span>)</span>,<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="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K13" title="SURREALI:func.13">divset</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="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="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="surreali.html#K6" title="SURREALI:func.6">divR</a> (<span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="surreali.html#T34" target="_self" onmouseover="rs('surreali/T34')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E9:56"/>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p2"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p3">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p4">(<span class="default"><a href="surreali.html#K5" title="SURREALI:func.5">divL</a> (<span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span>,<span class="p5">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p0"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span></span>)</span>,<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 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 class="p3">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p4"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="surreali.html#K3" title="SURREALI:func.3">divset</a> (<span class="p2">(<span class="default"><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>,<span class="p4">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p5"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span></span>)</span>,<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 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>,<span class="p2">(<span class="default"><a href="surreali.html#K11" title="SURREALI:func.11">No_inverses_on</a> <span class="p3"><a href="surreali.html#K7" title="SURREALI:func.7">||.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="surreali.html#K7" title="SURREALI:func.7">.||</a></span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surreali.html#T12" target="_self" onmouseover="rs('surreali/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:56"/>
<font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p5">(<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="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 class="p5">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<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 class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Maroon" title="c1">x</font></span>)</span>,<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 class="p4">(<span class="default"><a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="surreali.html#K13" title="SURREALI:func.13">divset</a> (<span class="p4">(<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="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 class="p4">(<span class="default"><a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></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="#E6:56"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="surreali.html#T13" target="_self" onmouseover="rs('surreali/T13')" onmouseout="rh()">Th13</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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