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<b>let </b><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">B1</font>, <font color="Maroon" title="c4">B2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c5">F</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D</font>;<br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:5"/><i><font color="Green" title="E5">A1</font></i>: 
( <font color="Maroon" title="c5">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon" title="c5">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )
 <b>and </b><br/><a NAME="E2:5"/><i><font color="Green" title="E6">A2</font></i>: 
( ( <font color="Maroon" title="c3">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">B2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or <font color="Maroon" title="c5">F</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  )
 <b>and </b><br/><a NAME="E3:5"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c4">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1"/>
( <font color="Maroon" title="c3">B1</font> <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="c4">B2</font> <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="c3">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">B2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1"><b>suppose </b></a><a NAME="E1:5_1_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
( <font color="Maroon" title="c3">B1</font> <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="c4">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1_1_1_1"/>
( <font color="Maroon" title="c3">B1</font> <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="c4">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span><b>by </b><i><a class="txt" href="setwop_2.html#E1:5_1_1_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:5_1_1_1_1_1_1"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c4">B2</font></span>)</span>,<font color="Maroon" title="c6">f</font> = 
<font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c5">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c4">B2</font>,<font color="Maroon" title="c6">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E2:5"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="setwiseo.html#T23" title="SETWISEO:th.23">SETWISEO:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c4">B2</font>,<font color="Maroon" title="c6">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="setwop_2.html#E2:5"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="setwop_2.html#E1:5_1_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="setwiseo.html#T40" title="SETWISEO:th.40">SETWISEO:40</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c4">B2</font>,<font color="Maroon" title="c6">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:5_1_1_1_1_1_1"><i><font color="Green" title="E9">A5</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:5_1_1_1_1_1_2"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c4">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c4">B2</font></span>)</span>,<font color="Maroon" title="c6">f</font> = 
<font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c5">F</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E2:5"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="setwiseo.html#T23" title="SETWISEO:th.23">SETWISEO:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c6">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="setwop_2.html#E2:5"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="setwop_2.html#E1:5_1_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="setwiseo.html#T40" title="SETWISEO:th.40">SETWISEO:40</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c4">B2</font>,<font color="Maroon" title="c6">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:5_1_1_1_1_1_2"><i><font color="Green" title="E9">A6</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:5_1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c4">B2</font></span>)</span>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c4">B2</font>,<font color="Maroon" title="c6">f</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_2"><b>suppose </b></a><a NAME="E1:5_1_1_2"/><i><font color="Green" title="E8">A7</font></i>: 
( <font color="Maroon" title="c3">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">B2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</font>] <b>means </b>( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">B1</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> $1</span>)</span>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> $1,<font color="Maroon" title="c6">f</font></span>)</span> );<br/><a NAME="E2:5_1_1_2"/><i><font color="Green" title="E9">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">C</font>]
 ;<br/><a NAME="E3:5_1_1_2"/><i><font color="Green" title="E10">A9</font></i>: 
for <font color="Olive" title="b1">B'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B'</font>] &amp; not <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B'</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B'</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c8">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:5_1_1_2_1"/><i><font color="Green" title="E11">A10</font></i>: 
( <font color="Maroon" title="c7">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">B1</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c6">f</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:5_1_1_2_1"/><i><font color="Green" title="E12">A11</font></i>: 
not <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <b>and </b><br/><a NAME="E3:5_1_1_2_1"/>
<font color="Maroon" title="c7">B</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="c8">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E4:5_1_1_2_1"/><i><font color="Green" title="E13">A12</font></i>: 
( <font color="Maroon" title="c7">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">B</font> implies <font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c6">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="setwop_2.html#E1:5_1_1_2_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>assume </b><a NAME="E5:5_1_1_2_1"/><i><font color="Green" title="E14">A13</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E6:5_1_1_2_1"/><b>then </b><i><font color="Green" title="E15">A14</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">B</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="c8">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E7:5_1_1_2_1"/>
( <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</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="c8">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="setwop_2.html#E5:5_1_1_2_1"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="setwiseo.html#T6" title="SETWISEO:th.6">SETWISEO:6</a></i>;<br/>
<a NAME="E8:5_1_1_2_1"/><b>then </b><i><font color="Green" title="E16">A15</font></i>: 
not <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1_2_1_1_1"/>
( <font color="Maroon" title="c7">B</font> <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="c7">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:5_1_1_2_1_1_1_1"/><i><font color="Green" title="E17">A16</font></i>: 
<font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c6">f</font> = 
<font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="setwop_2.html#E1:5_1_1_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="setwop_2.html#E8:5_1_1_2_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="setwop_2.html#T4" target="_self" title="SETWOP_2:th.4">Th4</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c6">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="setwop_2.html#E1:5_1_1_2_1_1_1_1"><i><font color="Green" title="E17">A16</font></i></a>, <a class="ref" href="setwiseo.html#T26" title="SETWISEO:th.26">SETWISEO:26</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:5_1_1_2_1_1_1_2"/><i><font color="Green" title="E17">A17</font></i>: 
<font color="Maroon" title="c7">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:5_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E18">A18</font></i>: 
( <font color="Maroon" title="c3">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; not <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c7">B</font> )
 <b>by </b><i><a class="txt" href="setwop_2.html#E2:5_1_1_2_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="setwop_2.html#E8:5_1_1_2_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a></i>;<br/><a NAME="E3:5_1_1_2_1_1_1_2"/><i><font color="Green" title="E19">A19</font></i>: 
( <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</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="c8">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">B1</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c3">B1</font> )
 <b>by </b><i><a class="txt" href="setwop_2.html#E5:5_1_1_2_1"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>thus </b><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c6">f</font> = 
<font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c6">f</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="setwop_2.html#E4:5_1_1_2_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="setwop_2.html#E6:5_1_1_2_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="setwop_2.html#E1:5_1_1_2_1_1_1_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="txt" href="setwop_2.html#E2:5_1_1_2_1_1_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="setwop_2.html#E3:5_1_1_2_1_1_1_2"><i><font color="Green" title="E19">A19</font></i></a>, <a class="ref" href="setwop_2.html#T4" target="_self" title="SETWOP_2:th.4">Th4</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c6">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="setwop_2.html#E2:5_1_1_2_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="setwop_2.html#E1:5_1_1_2_1_1_1_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="setwop_2.html#T4" target="_self" title="SETWOP_2:th.4">Th4</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:5_1_1_2_1"/>
<font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c6">f</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E4:5_1_1_2"/>
for <font color="Olive" title="b1">B2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B2</font>]
 <b>from </b><i><a class="ref" href="setwiseo.html#S2" title="SETWISEO:sch.2">SETWISEO:sch 2</a>(<a class="txt" href="setwop_2.html#E2:5_1_1_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="setwop_2.html#E3:5_1_1_2"><i><font color="Green" title="E10">A9</font></i></a>);<br/></i><b>hence </b><a NAME="E5:5_1_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c4">B2</font></span>)</span>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c4">B2</font>,<font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="setwop_2.html#E3:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="setwop_2.html#E1:5_1_1_2"><i><font color="Green" title="E8">A7</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:5"/>
<font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c4">B2</font></span>)</span>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c4">B2</font>,<font color="Maroon" title="c6">f</font></span>)</span>
 ;<br/>


</div>
