<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">g</font> be   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">g</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a>  iff  for <font color="Olive" title="b1">gR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c1">g</font> holds <br/> ex <font color="Olive" title="b2">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Olive" title="b1">gR</font> &amp; <font color="Olive" title="b2">gRL</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a>  ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">g</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 holds <br/> for <font color="Olive" title="b2">gR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b2">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Olive" title="b1">g</font> holds <br/> ex <font color="Olive" title="b3">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b3">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Olive" title="b2">gR</font> &amp; <font color="Olive" title="b3">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 );<br/>
<a NAME="E1:90"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
 for <font color="Olive" title="b1">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">alpha</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Olive" title="b2">alpha</font></span>)</span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">alpha</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">s</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">s</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Olive" title="b1">alpha</font></span>)</span>]</span><br/><span class="kw">let </span><font color="Maroon" title="c3">alpha</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">s</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">s</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c3">alpha</font></span>)</span>] )</span><br/>



<span class="kw">assume </span><a NAME="E1:90_1"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">s</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">s</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c3">alpha</font></span>)</span>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">g</font> be   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c3">alpha</font></span>)</span> implies  for <font color="Olive" title="b1">gR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c4">g</font> holds <br/> ex <font color="Olive" title="b2">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Olive" title="b1">gR</font> &amp; <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c3">alpha</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:90_1"/>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c3">alpha</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">gR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c4">g</font> holds <br/> ex <font color="Olive" title="b2">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Olive" title="b1">gR</font> &amp; <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c3">alpha</font></span>)</span> )</span><br/>

<a NAME="E3:90_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A3</font></span>: 
( <font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> &amp; <font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c3">alpha</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">gR</font> be   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c4">g</font> implies  ex <font color="Olive" title="b1">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c5">gR</font> &amp; <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c3">alpha</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:90_1"/><span class="lab"><font color="Green" title="E60">A4</font></span>: 
<font color="Maroon" title="c5">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c4">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c5">gR</font> &amp; <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c3">alpha</font></span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a><span class="kw"> such that </span><br/><a NAME="E6:90_1"/><span class="lab"><font color="Green" title="E61">A5</font></span>: 
( <font color="Maroon" title="c6">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c5">gR</font> &amp; <font color="Maroon" title="c6">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E3:90_1"><span class="lab"><font color="Green" title="E59">A3</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">gRL</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c5">gR</font> &amp; <font color="Maroon" title="c6">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c3">alpha</font></span>)</span> )</span><br/>

<a NAME="E7:90_1"/>
<font color="Maroon" title="c5">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c3">alpha</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cgames_1.html#T11" target="_self" title="CGAMES_1:th.11">Th11</a>, <a class="txt" href="#E3:90_1"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="txt" href="#E4:90_1"><span class="lab"><font color="Green" title="E60">A4</font></span></a></span>;<br/>
<a NAME="E8:90_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c3">alpha</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cgames_1.html#T11" target="_self" title="CGAMES_1:th.11">Th11</a>, <a class="txt" href="#E6:90_1"><span class="lab"><font color="Green" title="E61">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:90_1"/>
( <font color="Maroon" title="c6">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c5">gR</font> &amp; <font color="Maroon" title="c6">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c3">alpha</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90_1"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" 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">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">gR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c1">g</font> holds <br/> ex <font color="Olive" title="b2">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Olive" title="b1">gR</font> &amp; <font color="Olive" title="b2">gRL</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a>  ) ) implies <font color="Maroon" title="c1">g</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:90_2"/>
<font color="Maroon" title="c1">g</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">gR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c1">g</font> holds <br/> ex <font color="Olive" title="b2">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Olive" title="b1">gR</font> &amp; <font color="Olive" title="b2">gRL</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a>  )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:90_2"/><span class="lab"><font color="Green" title="E58">A6</font></span>: 
( <font color="Maroon" title="c1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">s</font>] )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cgames_1.html#D15" target="_self" title="CGAMES_1:def.15">Def15</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c3">gR</font> be   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c1">g</font> implies  ex <font color="Olive" title="b1">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c3">gR</font> &amp; <font color="Olive" title="b1">gRL</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a>  ) )</span><br/><span class="kw">assume </span><a NAME="E4:90_2"/>
<font color="Maroon" title="c3">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c1">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c3">gR</font> &amp; <font color="Olive" title="b1">gRL</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a>  )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a><span class="kw"> such that </span><br/><a NAME="E6:90_2"/><span class="lab"><font color="Green" title="E59">A7</font></span>: 
( <font color="Maroon" title="c4">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c3">gR</font> &amp; <font color="Maroon" title="c4">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_2"><span class="lab"><font color="Green" title="E58">A6</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c5">gRL</font> = <font color="Maroon" title="c4">gRL</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c3">gR</font> &amp; <font color="Maroon" title="c5">gRL</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a>  )</span><br/><span class="kw">thus </span><a NAME="E7:90_2"/>
<font color="Maroon" title="c5">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c3">gR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90_2"><span class="lab"><font color="Green" title="E59">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">gRL</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a> </span><br/><span class="kw">thus </span><a NAME="E8:90_2"/>
<font color="Maroon" title="c5">gRL</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c2">s</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cgames_1.html#D15" target="_self" title="CGAMES_1:def.15">CGAMES_1:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> &amp; (  for <font color="Olive" title="b1">g1</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> holds <br/> for <font color="Olive" title="b2">gR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b2">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Olive" title="b1">g1</font> holds <br/> ex <font color="Olive" title="b3">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b3">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Olive" title="b2">gR</font> &amp; <font color="Olive" title="b3">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:90_2_1"/>
( <font color="Maroon" title="c5">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> &amp; (  for <font color="Olive" title="b1">g1</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> holds <br/> for <font color="Olive" title="b2">gR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b2">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Olive" title="b1">g1</font> holds <br/> ex <font color="Olive" title="b3">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b3">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Olive" title="b2">gR</font> &amp; <font color="Olive" title="b3">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">s</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_2"><span class="lab"><font color="Green" title="E58">A6</font></span></a>, <a class="txt" href="#E6:90_2"><span class="lab"><font color="Green" title="E59">A7</font></span></a></span>; <a class="txt" 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>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:90_3"/><span class="lab"><font color="Green" title="E58">A8</font></span>: 
 for <font color="Olive" title="b1">gR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c1">g</font> holds <br/> ex <font color="Olive" title="b2">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Olive" title="b1">gR</font> &amp; <font color="Olive" title="b2">gRL</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">g</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c2">alpha</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:90_3"/><span class="lab"><font color="Green" title="E59">A9</font></span>: 
<font color="Maroon" title="c1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cgames_1.html#D3" target="_self" title="CGAMES_1:def.3">Def3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">ss</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s1</font> where <font color="Olive" title="b1">s1</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s1</font>] </span> } </span> ;<br/><span class="kw">take </span><font color="Maroon" title="c4">s</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s1</font> where <font color="Olive" title="b1">s1</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s1</font>] </span> } </span> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">s</font>] )</span><br/><a NAME="E1:90_3_1"/><span class="lab"><font color="Green" title="E60">A10</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">s</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">g1</font> be   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font> implies  for <font color="Olive" title="b1">gR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c5">g1</font> holds <br/> ex <font color="Olive" title="b2">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Olive" title="b1">gR</font> &amp; <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:90_3_1_1"/>
<font color="Maroon" title="c5">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">gR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c5">g1</font> holds <br/> ex <font color="Olive" title="b2">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Olive" title="b1">gR</font> &amp; <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">s2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:90_3_1_1"/><span class="lab"><font color="Green" title="E61">A11</font></span>: 
( <font color="Maroon" title="c5">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">s2</font> &amp; <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s1</font> where <font color="Olive" title="b1">s1</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s1</font>] </span> } </span>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">s1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:90_3_1_1"/><span class="lab"><font color="Green" title="E62">A12</font></span>: 
( <font color="Maroon" title="c7">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">s1</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_3_1_1"><span class="lab"><font color="Green" title="E61">A11</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">gR</font> be   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c5">g1</font> implies  ex <font color="Olive" title="b1">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c8">gR</font> &amp; <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E6:90_3_1_1"/>
<font color="Maroon" title="c8">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c5">g1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c8">gR</font> &amp; <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a><span class="kw"> such that </span><br/><a NAME="E8:90_3_1_1"/><span class="lab"><font color="Green" title="E63">A13</font></span>: 
( <font color="Maroon" title="c9">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c8">gR</font> &amp; <font color="Maroon" title="c9">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">s1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_3_1_1"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="txt" href="#E5:90_3_1_1"><span class="lab"><font color="Green" title="E62">A12</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">gRL</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c8">gR</font> &amp; <font color="Maroon" title="c9">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font> )</span><br/>

<a NAME="E9:90_3_1_1"/>
<font color="Maroon" title="c6">s2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_3_1_1"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:90_3_1_1"/>
( <font color="Maroon" title="c9">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c8">gR</font> &amp; <font color="Maroon" title="c9">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90_3_1_1"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="txt" href="#E8:90_3_1_1"><span class="lab"><font color="Green" title="E63">A13</font></span></a></span>; <a class="txt" 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">thus </span><a NAME="E2:90_3_1"/>
<font color="Maroon" title="c1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">s</font>]</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_3_1_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_3_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s1</font> where <font color="Olive" title="b1">s1</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s1</font>] </span> } </span>  implies <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font> )</span><br/><span class="kw">assume </span><a NAME="E1:90_3_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s1</font> where <font color="Olive" title="b1">s1</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s1</font>] </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font></span><br/><a NAME="E2:90_3_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font></span>)</span> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s1</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s1</font>] )
 ;<br/><span class="kw">hence </span><a NAME="E3:90_3_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font>
 ; <a class="txt" 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>
<a NAME="E2:90_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A14</font></span>: 
<font color="Maroon" title="c4">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T76" title="ZFMISC_1:th.76">ZFMISC_1:76</a></span>;<br/>
<a NAME="E3:90_3_1_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">g</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_3"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">sg</font> = <font color="Maroon" title="c4">s</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">g</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:90_3_1_2"><span class="lab"><font color="Green" title="E61">A14</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E5:90_3_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">sg</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_3_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">g1</font> be   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font> implies  for <font color="Olive" title="b1">gR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c6">g1</font> holds <br/> ex <font color="Olive" title="b2">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Olive" title="b1">gR</font> &amp; <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:90_3_1_2_2"/><span class="lab"><font color="Green" title="E62">A15</font></span>: 
<font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">gR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c6">g1</font> holds <br/> ex <font color="Olive" title="b2">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Olive" title="b1">gR</font> &amp; <font color="Olive" title="b2">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">gR</font> be   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c6">g1</font> implies  ex <font color="Olive" title="b1">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c7">gR</font> &amp; <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:90_3_1_2_2"/><span class="lab"><font color="Green" title="E63">A16</font></span>: 
<font color="Maroon" title="c7">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K7" title="CGAMES_1:func.7">the_RightOptions_of</a> <font color="Maroon" title="c6">g1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c7">gR</font> &amp; <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_3_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:90_3_1_2_2_1"/>
( <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font> or <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">g</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_2_2"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_3_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:90_3_1_2_2_1_1"/>
<font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c7">gR</font> &amp; <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a><span class="kw"> such that </span><br/><a NAME="E3:90_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E64">A17</font></span>: 
( <font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c7">gR</font> &amp; <font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="txt" href="#E2:90_3_1_2_2"><span class="lab"><font color="Green" title="E63">A16</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c8">gRL</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c7">gR</font> &amp; <font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font> )</span><br/><span class="kw">thus </span><a NAME="E4:90_3_1_2_2_1_1"/>
<font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c7">gR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E64">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font></span><br/><a NAME="E5:90_3_1_2_2_1_1"/>
<font color="Maroon" title="c4">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">sg</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:90_3_1_2_2_1_1"/>
<font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E64">A17</font></span></a></span>; <a class="txt" 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="90_3_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:90_3_1_2_2_1_2"/>
<font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">g</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c7">gR</font> &amp; <font color="Olive" title="b1">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font> )</span><br/><div class="add"><a NAME="E2:90_3_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">gRL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a><span class="kw"> such that </span><br/><a NAME="E4:90_3_1_2_2_1_2"/><span class="lab"><font color="Green" title="E64">A18</font></span>: 
( <font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c7">gR</font> &amp; <font color="Maroon" title="c8">gRL</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="txt" href="#E2:90_3_1_2_2"><span class="lab"><font color="Green" title="E63">A16</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">s0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:90_3_1_2_2_1_2"/><span class="lab"><font color="Green" title="E65">A19</font></span>: 
( <font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">s0</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">s0</font>] )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cgames_1.html#D15" target="_self" title="CGAMES_1:def.15">Def15</a>, <a class="txt" href="#E4:90_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E64">A18</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c8">gRL</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c7">gR</font> &amp; <font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font> )</span><br/><span class="kw">thus </span><a NAME="E7:90_3_1_2_2_1_2"/>
<font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K6" title="CGAMES_1:func.6">the_LeftOptions_of</a> <font color="Maroon" title="c7">gR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E64">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c10">s1</font> = <font color="Maroon" title="c9">s0</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><a NAME="E9:90_3_1_2_2_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">s1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E65">A19</font></span></a>, <a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E57">A1</font></span></a></span>;<br/><a NAME="E10:90_3_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s1</font> where <font color="Olive" title="b1">s1</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s1</font>] </span> } </span> 
 ;<br/><a NAME="E11:90_3_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A20</font></span>: 
<font color="Maroon" title="c10">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/><a NAME="E12:90_3_1_2_2_1_2"/>
<font color="Maroon" title="c7">gR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1_2_2"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="txt" href="#E2:90_3_1_2_2"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="ref" href="cgames_1.html#T11" target="_self" title="CGAMES_1:th.11">Th11</a></span>;<br/><a NAME="E13:90_3_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E64">A18</font></span></a>, <a class="ref" href="cgames_1.html#T11" target="_self" title="CGAMES_1:th.11">Th11</a></span>;<br/><a NAME="E14:90_3_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E65">A19</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E15:90_3_1_2_2_1_2"/>
<font color="Maroon" title="c8">gRL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:90_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E66">A20</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" 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>

</div><span class="kw">end;</span></div>
<a NAME="E6:90_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A21</font></span>: 
<font color="Maroon" title="c5">sg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s1</font> where <font color="Olive" title="b1">s1</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="cgames_1.html#K2" title="CGAMES_1:func.2">ConwayDay</a> <font color="Maroon" title="c2">alpha</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s1</font>] </span> } </span> 
 
;<br/>
<a NAME="E7:90_3_1_2"/>
<font color="Maroon" title="c1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">g</font></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" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E8:90_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">sg</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:90_3_1_2"/>
<font color="Maroon" title="c1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90_3_1_2"><span class="lab"><font color="Green" title="E62">A21</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" 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">thus </span><a NAME="E3:90_3_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">s</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3_1"><span class="lab"><font color="Green" title="E60">A10</font></span></a></span>; <a class="txt" 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="E5:90_3"/>
<font color="Maroon" title="c1">g</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cgames_1.html#D15" target="_self" title="CGAMES_1:def.15">Def15</a></span>; <a class="txt" 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>
