<?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#V6" title="CGAMES_1:attr.6">nonpositive</a>  iff  for <font color="Olive" title="b1">gL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">gL</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="c1">g</font> holds <br/> ex <font color="Olive" title="b2">gLR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b2">gLR</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">gL</font> &amp; <font color="Olive" title="b2">gLR</font> is  <a href="cgames_1.html#V6" title="CGAMES_1:attr.6">nonpositive</a>  ) )</span><br/>

<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">gL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">gL</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="c1">g</font> holds <br/> ex <font color="Olive" title="b2">gLR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b2">gLR</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">gL</font> &amp; <font color="Olive" title="b2">gLR</font> is  <a href="cgames_1.html#V6" title="CGAMES_1:attr.6">nonpositive</a>  ) ) implies <font color="Maroon" title="c1">g</font> is  <a href="cgames_1.html#V6" title="CGAMES_1:attr.6">nonpositive</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:91_1"/>
<font color="Maroon" title="c1">g</font> is  <a href="cgames_1.html#V6" title="CGAMES_1:attr.6">nonpositive</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">gL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">gL</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="c1">g</font> holds <br/> ex <font color="Olive" title="b2">gLR</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">gLR</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">gL</font> &amp; <font color="Olive" title="b2">gLR</font> is  <a href="cgames_1.html#V6" title="CGAMES_1:attr.6">nonpositive</a>  )</span><br/><a NAME="E2:91_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A1</font></span>: 
 <a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <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#D16" target="_self" title="CGAMES_1:def.16">Def16</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c2">gL</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="c2">gL</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="c1">g</font> implies  ex <font color="Olive" title="b1">gLR</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">gLR</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="c2">gL</font> &amp; <font color="Olive" title="b1">gLR</font> is  <a href="cgames_1.html#V6" title="CGAMES_1:attr.6">nonpositive</a>  ) )</span><br/><span class="kw">assume </span><a NAME="E3:91_1"/>
<font color="Maroon" title="c2">gL</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="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">gLR</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">gLR</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="c2">gL</font> &amp; <font color="Olive" title="b1">gLR</font> is  <a href="cgames_1.html#V6" title="CGAMES_1:attr.6">nonpositive</a>  )</span><br/><a NAME="E4:91_1"/><span class="kw">then </span>
 <a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c2">gL</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> <span class="p1">(<span class="default"><a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c1">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cgames_1.html#T39" target="_self" title="CGAMES_1:th.39">Th39</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">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="E6:91_1"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
( <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c2">gL</font></span>)</span> &amp; <font color="Maroon" title="c3">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="#E2:91_1"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="cgames_1.html#T43" target="_self" title="CGAMES_1:th.43">Th43</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c4">gLR</font> =  <a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c3">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="c4">gLR</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="c2">gL</font> &amp; <font color="Maroon" title="c4">gLR</font> is  <a href="cgames_1.html#V6" title="CGAMES_1:attr.6">nonpositive</a>  )</span><br/><a NAME="E7:91_1"/>
( <font color="Maroon" title="c4">gLR</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> <span class="p1">(<span class="default"><a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <span class="p2">(<span class="default"><a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c2">gL</font></span>)</span></span>)</span> &amp;  <a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c2">gL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">gL</font> &amp;  <a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c4">gLR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">gRL</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:91_1"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="cgames_1.html#T39" target="_self" title="CGAMES_1:th.39">Th39</a>, <a class="ref" href="cgames_1.html#T40" target="_self" title="CGAMES_1:th.40">Th40</a></span>;<br/><span class="kw">hence </span><a NAME="E8:91_1"/>
( <font color="Maroon" title="c4">gLR</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="c2">gL</font> &amp; <font color="Maroon" title="c4">gLR</font> is  <a href="cgames_1.html#V6" title="CGAMES_1:attr.6">nonpositive</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cgames_1.html#D16" target="_self" title="CGAMES_1:def.16">Def16</a>, <a class="txt" href="#E6:91_1"><span class="lab"><font color="Green" title="E59">A2</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">assume </span><a NAME="E2:91"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
 for <font color="Olive" title="b1">gL</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a>  st <font color="Olive" title="b1">gL</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="c1">g</font> holds <br/> ex <font color="Olive" title="b2">gLR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a> st <br/>( <font color="Olive" title="b2">gLR</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">gL</font> &amp; <font color="Olive" title="b2">gLR</font> is  <a href="cgames_1.html#V6" title="CGAMES_1:attr.6">nonpositive</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#V6" title="CGAMES_1:attr.6">nonpositive</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">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="c2">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> <span class="p1">(<span class="default"><a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c1">g</font></span>)</span> implies  ex <font color="Olive" title="b1">gRL</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c2">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="E1:91_2"/>
<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c1">g</font></span>)</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="hidden.html#M1" title="HIDDEN:mode.1">set</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="c2">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/><a NAME="E2:91_2"/><span class="kw">then </span>
(  <a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c2">gR</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> <span class="p1">(<span class="default"><a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <span class="p2">(<span class="default"><a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span> &amp;  <a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c1">g</font></span>)</span> <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="cgames_1.html#T39" target="_self" title="CGAMES_1:th.39">Th39</a>, <a class="ref" href="cgames_1.html#T40" target="_self" title="CGAMES_1:th.40">Th40</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">gLR</font> being   <a href="cgames_1.html#NM1" title="CGAMES_1:NM.1">ConwayGame</a><span class="kw"> such that </span><br/><a NAME="E4:91_2"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
( <font color="Maroon" title="c3">gLR</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> <span class="p1">(<span class="default"><a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c2">gR</font></span>)</span> &amp; <font color="Maroon" title="c3">gLR</font> is  <a href="cgames_1.html#V6" title="CGAMES_1:attr.6">nonpositive</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91"><span class="lab"><font color="Green" title="E58">A3</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c4">gRL</font> =  <a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c3">gLR</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="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="c2">gR</font> &amp; <font color="Maroon" title="c4">gRL</font> is  <a href="cgames_1.html#V5" title="CGAMES_1:attr.5">nonnegative</a>  )</span><br/><a NAME="E5:91_2"/>
( <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> <span class="p1">(<span class="default"><a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <span class="p2">(<span class="default"><a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c2">gR</font></span>)</span></span>)</span> &amp;  <a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <span class="p1">(<span class="default"><a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <font color="Maroon" title="c2">gR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">gR</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91_2"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="cgames_1.html#T39" target="_self" title="CGAMES_1:th.39">Th39</a>, <a class="ref" href="cgames_1.html#T40" target="_self" title="CGAMES_1:th.40">Th40</a></span>;<br/><span class="kw">hence </span><a NAME="E6:91_2"/>
( <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="c2">gR</font> &amp; <font color="Maroon" title="c4">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="#E4:91_2"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="cgames_1.html#D16" target="_self" title="CGAMES_1:def.16">Def16</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>
<a NAME="E4:91"/><span class="kw">then </span>
 <a href="cgames_1.html#K13" title="CGAMES_1:func.13">-</a> <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#T43" target="_self" title="CGAMES_1:th.43">Th43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:91"/>
<font color="Maroon" title="c1">g</font> is  <a href="cgames_1.html#V6" title="CGAMES_1:attr.6">nonpositive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cgames_1.html#D16" target="_self" title="CGAMES_1:def.16">Def16</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>
