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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>$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#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">$1 <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c1">lamb</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Inv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<a NAME="E1:2_1_1"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</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">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font>
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">D</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1"/><span class="lab"><font color="Green" title="E3">A2</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="c5">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> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &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="#E1:2_1_1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c5">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="c5">D</font> iff  ex <font color="Olive" title="b2">xL</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">xL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b2">xL</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &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#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">xL</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c1">lamb</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Inv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">xL</font></span>)</span> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">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="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> iff  ex <font color="Olive" title="b1">xL</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">xL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">xL</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <font color="Maroon" title="c6">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#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">xL</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c1">lamb</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Inv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">xL</font></span>)</span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:2_1_1"/>
( <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> implies  ex <font color="Olive" title="b1">xL</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">xL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">xL</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <font color="Maroon" title="c6">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#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">xL</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c1">lamb</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Inv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">xL</font></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">xL</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">xL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">xL</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <font color="Maroon" title="c6">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#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">xL</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c1">lamb</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Inv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">xL</font></span>)</span> ) implies <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_1_1_1"/>
<font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">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">xL</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">xL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">xL</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <font color="Maroon" title="c6">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#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">xL</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c1">lamb</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Inv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">xL</font></span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_1"/><span class="lab"><font color="Green" title="E4">A3</font></span>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">o</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a></span>;<br/>
<a NAME="E4:2_1_1_1"/>
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:2_1_1_1"/>
 ex <font color="Olive" title="b1">xL</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">xL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">xL</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  &amp; <font color="Maroon" title="c6">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#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">xL</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c1">lamb</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Inv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">xL</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_1"><span class="lab"><font color="Green" title="E4">A3</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">given </span><font color="Maroon" title="c7">xL</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E6:2_1_1"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
( <font color="Maroon" title="c7">xL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c7">xL</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a>  )
 <span class="kw">and </span><br/><a NAME="E7:2_1_1"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
<font color="Maroon" title="c6">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#K12" title="SURREALR:func.12">+'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">xL</font> <a href="surrealr.html#K12" title="SURREALR:func.12">+'</a> <span class="p4">(<span class="default"><a href="surrealr.html#K11" title="SURREALR:func.11">-'</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <font color="Maroon" title="c1">lamb</font></span>)</span></span>)</span> <a href="surreali.html#K1" title="SURREALI:func.1">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Inv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">xL</font></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="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font></span><br/>

<a NAME="E8:2_1_1"/>
<font color="Maroon" title="c7">xL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="surreal0.html#K11" title="SURREAL0:func.11">0_No</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:2_1_1"/>
<font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E7:2_1_1"><span class="lab"><font color="Green" title="E5">A5</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>
