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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"> (  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>  implies <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:51"/><span class="lab"><font color="Green" title="E43">A1</font></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> 
 ; <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#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1"/>
(  <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="c1">x</font> or  not  <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="c1">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><span class="kw">suppose </span></a><a NAME="E1:51_1_1"/>
 <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> </span><br/><div class="add"><a NAME="E2:51_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 ;<br/><a NAME="E3:51_1_1"/><span class="kw">then </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="#E1:51"><span class="lab"><font color="Green" title="E43">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">hence </span><a NAME="E4:51_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="surreali.html#E45"><span class="lab"><font color="Green" title="E40">Lm2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_2"><span class="kw">suppose </span></a><a NAME="E1:51_1_2"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
 not  <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="c1">x</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> </span><br/><div class="add"><a NAME="E2:51_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A4</font></span>: 
 <a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="surrealr.html#T23" onmouseover="rs('surrealr/T23')" onmouseout="rh()">SURREALR:23</a>, <a class="ref" href="surrealr.html#T10" onmouseover="rs('surrealr/T10')" onmouseout="rh()">SURREALR:10</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">I</font> =  <a href="surreali.html#K10" title="SURREALI:func.10">inv</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> ;<br/><a NAME="E4:51_1_2"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
 not <font color="Maroon" title="c1">x</font> is  <a href="surreali.html#V2" title="SURREALI:attr.2">positive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_2"><span class="lab"><font color="Green" title="E44">A3</font></span></a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c1">x</font> = 
<span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</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="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealr.html#T58" onmouseover="rs('surrealr/T58')" onmouseout="rh()">SURREALR:58</a></span>
<br/>.= 
<font color="Maroon" title="c2">I</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1_2"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E43">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">hence </span><a NAME="E6:51_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1_2"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="surreali.html#E45"><span class="lab"><font color="Green" title="E40">Lm2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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