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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">n</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font> &amp;  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font> &amp;  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> )</span><br/>



<a NAME="E1:102"/>
<font color="Maroon" title="c1">x</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span> 
 
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<a NAME="E2:102"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D8" target="_self" onmouseover="rs('surrealn/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<a NAME="E3:102"/>
<font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span> 
 
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<a NAME="E4:102"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A2</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D8" target="_self" onmouseover="rs('surrealn/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<a NAME="E5:102"/>
(  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="surreal0.html#R7" title="SURREAL0:pred.7">&lt;&lt;</a>  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp;  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="surrealo.html#T11" onmouseover="rs('surrealo/T11')" onmouseout="rh()">SURREALO:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:102"/>
( <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a>  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font> &amp;  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:102"><span class="lab"><font color="Green" title="E83">A1</font></span></a>, <a class="txt" href="#E4:102"><span class="lab"><font color="Green" title="E84">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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