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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span>( $1 is  <a href="xtuple_0.html#V1" title="XTUPLE_0:attr.1">pair</a>  &amp;  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> $1 is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  &amp;  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> $1 is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  &amp; (  for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>(  <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="Olive" title="b1">x1</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b2">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> ) ) );<br/>
<a NAME="E1:52_1_1"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>] holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</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">S<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>] implies <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:52_1_1_1"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>] )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:52_1_1_1"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#D1" onmouseover="rs('xtuple_0/D1')" onmouseout="rh()">XTUPLE_0:def 1</a>, <a class="txt" href="#E1:52_1_1_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">a1</font> = <font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c10">a2</font> = <font color="Maroon" title="c8">a2</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1_1_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E1:52_1_1_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c5">b</font> = 
<span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c10">a2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1_1_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E1:52_1_1_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1_1_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E1:52_1_1_1"><span class="lab"><font color="Green" title="E44">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/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c4">D</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:52_1_1"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> iff  ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="tarski.html#S1" onmouseover="rs('tarski/S1')" onmouseout="rh()">TARSKI:sch 1</a>(<a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E43">A1</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c4">D</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">  for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> iff  ex <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> st <br/>(  <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="Olive" title="b2">x1</font> &amp; <font color="Olive" title="b2">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b3">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b3">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">o</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="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> iff  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> st <br/>(  <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="Olive" title="b1">x1</font> &amp; <font color="Olive" title="b1">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b2">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:52_1_1"/>
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> implies  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> st <br/>(  <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="Olive" title="b1">x1</font> &amp; <font color="Olive" title="b1">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b2">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> ) )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> st <br/>(  <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="Olive" title="b1">x1</font> &amp; <font color="Olive" title="b1">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b2">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> ) implies <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="52_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:52_1_1_2"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</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">  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> st <br/>(  <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="Olive" title="b1">x1</font> &amp; <font color="Olive" title="b1">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b2">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:52_1_1_2"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">o</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:52_1_1_2"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#D1" onmouseover="rs('xtuple_0/D1')" onmouseout="rh()">XTUPLE_0:def 1</a>, <a class="txt" href="#E3:52_1_1_2"><span class="lab"><font color="Green" title="E45">A5</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">a1</font> = <font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c10">a2</font> = <font color="Maroon" title="c8">a2</font> as   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52_1_1_2"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E3:52_1_1_2"><span class="lab"><font color="Green" title="E45">A5</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">a1</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">  ex <font color="Olive" title="b1">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a> st <br/>(  <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="c9">a1</font> &amp; <font color="Maroon" title="c9">a1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b1">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c10">a2</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c9">a1</font> &amp; <font color="Maroon" title="c9">a1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c10">a2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c10">a2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E7:52_1_1_2"/>
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c9">a1</font> &amp; <font color="Maroon" title="c9">a1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c10">a2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c10">a2</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52_1_1_2"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E3:52_1_1_2"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">given </span><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a><span class="kw"> such that </span><a NAME="E6:52_1_1"/><span class="lab"><font color="Green" title="E45">A7</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="c6">x1</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c7">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 <span class="kw">and </span><br/><a NAME="E7:52_1_1"/><span class="lab"><font color="Green" title="E46">A8</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="surrealr.html#K10" title="SURREALR:func.10">-</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c7">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font></span><br/>

<a NAME="E8:52_1_1"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52_1_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E9:52_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c5">o</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="52_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:52_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is  <a href="xtuple_0.html#V1" title="XTUPLE_0:attr.1">pair</a>  &amp;  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  &amp;  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is  <a href="surreal0.html#V2" title="SURREAL0:attr.2">surreal</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>(  <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="Olive" title="b1">x1</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Olive" title="b2">y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">X1</font>, <font color="Maroon" title="c9">Y1</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">X1</font>,<font color="Maroon" title="c9">Y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies (  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c8">X1</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">X1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">Y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">X1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:52_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">X1</font>,<font color="Maroon" title="c9">Y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c8">X1</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">X1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">Y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">X1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> )</span><br/>

<a NAME="E3:52_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">X1</font> &amp; <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Y1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:52_1_1_3"/>
(  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c8">X1</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surrealo.html#K1" title="SURREALO:func.1">1_No</a> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">X1</font> <a href="surrealr.html#K8" title="SURREALR:func.8">+</a> <span class="p4">(<span class="default"><a href="surrealr.html#K2" title="SURREALR:func.2">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <font color="Maroon" title="c9">Y1</font></span>)</span></span>)</span> <a href="surrealr.html#K16" title="SURREALR:func.16">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">X1</font> <a href="surreali.html#K12" title="SURREALI:func.12">"</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52_1_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="txt" href="#E7:52_1_1"><span class="lab"><font color="Green" title="E46">A8</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:52_1_1"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:52_1_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="txt" href="#E3:52_1_1"><span class="lab"><font color="Green" title="E44">A4</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>
