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

<span class="kw">set </span><font color="Maroon" title="c1">B</font> =  <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">X</font> =  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a> ;<br/>
<a NAME="E1:37"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">c</font> = <font color="Maroon" title="c2">a</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of 3 <span class="kw">by</span> <span class="lab"><a class="ref" href="robbins4.html#T10" target="_self" title="ROBBINS4:th.10">Th10</a></span>;<br/>
<a NAME="E3:37"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">c</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins4.html#D4" target="_self" title="ROBBINS4:def.4">Def4</a></span>;<br/>
<a NAME="E4:37"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins4.html#D4" target="_self" title="ROBBINS4:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:37"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )</span><br/>

<a NAME="E6:37"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
( <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="yellow11.html#T1" title="YELLOW11:th.1">YELLOW11:1</a>, <a class="ref" href="zfmisc_1.html#T142" title="ZFMISC_1:th.142">ZFMISC_1:142</a></span>;<br/>
<span class="kw">thus </span><a NAME="E7:37"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:37_1"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:37_1"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="yellow11.html#T1" title="YELLOW11:th.1">YELLOW11:1</a></span>;<br/>
<a NAME="E3:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A5</font></span>: 
not 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E4:37_1"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="yellow11.html#T1" title="YELLOW11:th.1">YELLOW11:1</a></span>;<br/>
<a NAME="E5:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A6</font></span>: 
not 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E6:37_1"/>
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E7:37_1"/><span class="kw">then </span>
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins4.html#D4" target="_self" title="ROBBINS4:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:37_1"/>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E5:37_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E8:37"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E30">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )</span><br/>

<a NAME="E9:37"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="yellow11.html#T1" title="YELLOW11:th.1">YELLOW11:1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E10:37"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 ) &amp; ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:37_2"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E2:37_2"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="yellow11.html#T3" title="YELLOW11:th.3">YELLOW11:3</a></span>;<br/>
<a NAME="E3:37_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A9</font></span>: 
not 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E4:37_2"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="yellow11.html#T3" title="YELLOW11:th.3">YELLOW11:3</a></span>;<br/>
<a NAME="E5:37_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A10</font></span>: 
not 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E6:37_2"/>
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="yellow11.html#T3" title="YELLOW11:th.3">YELLOW11:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:37_2"/>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E6:37"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E3:37_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E5:37_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E11:37"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E30">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 implies <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><br/>

<span class="kw">assume </span><a NAME="E12:37"/><span class="lab"><font color="Green" title="E33">A11</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>

<a NAME="E13:37"/><span class="kw">then </span>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="yellow11.html#T4" title="YELLOW11:th.4">YELLOW11:4</a></span>;<br/>
<a NAME="E14:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A12</font></span>: 
not 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E15:37"/>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 3 &amp; not 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="yellow11.html#T1" title="YELLOW11:th.1">YELLOW11:1</a>, <a class="ref" href="yellow11.html#T4" title="YELLOW11:th.4">YELLOW11:4</a></span>;<br/>
<a NAME="E16:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A13</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E17:37"/>
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="yellow11.html#T4" title="YELLOW11:th.4">YELLOW11:4</a></span>;<br/>
<a NAME="E18:37"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:37"/>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E14:37"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E16:37"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="ref" href="card_1.html#T88" title="CARD_1:th.88">CARD_1:88</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
