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<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 <font color="Maroon">F<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">F<sub>1</sub></font>()] <b>means </b> ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,$1] &amp; ( for <font color="Olive" title="b2">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">z</font>,<font color="Olive" title="b1">y</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font>] ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</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">F<sub>1</sub></font>()] <b>means </b>for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,$1];<br/>
<a NAME="E1:37_1"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon">F<sub>1</sub></font>()]
 
;<br/>
<div><i><font color="Green" title="E13">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">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">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">B</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">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="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">B</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">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="Maroon" title="c2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>] )</span><br/><b>assume </b><a NAME="E1:37_1_1"/><i><font color="Green" title="E14">A5</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">B</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">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="Maroon" title="c2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/><b>thus </b><a NAME="E2:37_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">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="Maroon" title="c2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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="Maroon" title="c2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c1">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="Maroon" title="c2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>] )</span><br/>

<b>assume </b><a NAME="E1:37_1_1_1"/><i><font color="Green" title="E15">A6</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c1">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="Maroon" title="c2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_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="E1:37_1_1_1_1_1"/>
( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> or <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 </span><b>by </b><i><a class="txt" href="fraenkel.html#E1:37_1_1_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="setwiseo.html#T6" title="SETWISEO:th.6">SETWISEO:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c1">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="Maroon" title="c2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:37_1_1_1_1_1_1"/><i><font color="Green" title="E16">A7</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 <b>and </b><br/><a NAME="E4:37_1_1_1_1_1_1"/><i><font color="Green" title="E17">A8</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">z</font>]
 <b>and </b><br/><a NAME="E5:37_1_1_1_1_1_1"/><i><font color="Green" title="E18">A9</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">y</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Olive" title="b1">x</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:37_1_1"><i><font color="Green" title="E14">A5</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_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="E1:37_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>] or  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:37_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E19">A10</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">z</font>] &amp; ( for <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c2">x</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">v</font>] ) )</span><br/><div class="add"><b>thus </b><a NAME="E2:37_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="setwiseo.html#T6" title="SETWISEO:th.6">SETWISEO:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">z</font>] &amp; ( for <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c2">x</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">v</font>] ) )</span><br/><b>thus </b><a NAME="E3:37_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">z</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E2:37"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="fraenkel.html#E4:37_1_1_1_1_1_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E19">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c2">x</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">v</font>]</span><br/><b>let </b><font color="Maroon" title="c5">v</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c2">x</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">v</font>] )</span><br/><b>assume </b><b>that </b><br/><a NAME="E4:37_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E20">A11</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E5:37_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E21">A12</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c2">x</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">v</font>]</span><br/><a NAME="E6:37_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E22">A13</font></i>: 
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 <b>by </b><i><a class="txt" href="fraenkel.html#E4:37_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="setwiseo.html#T6" title="SETWISEO:th.6">SETWISEO:6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E23">A14</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">v</font>]</span><br/><a NAME="E2:37_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c4">y</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E2:37"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="fraenkel.html#E5:37_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E21">A12</font></i></a></i>;<br/><a NAME="E3:37_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">v</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E5:37_1_1_1_1_1_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E23">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E4:37_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">v</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E2:37"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E19">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E8:37_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">v</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:37"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="fraenkel.html#E6:37_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E22">A13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:37_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E19">A15</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">z</font>] &amp; ( for <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c4">y</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Olive" title="b1">v</font>] ) )</span><br/><div class="add"><b>thus </b><a NAME="E2:37_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="fraenkel.html#E3:37_1_1_1_1_1_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">z</font>] &amp; ( for <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c4">y</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Olive" title="b1">v</font>] ) )</span><br/><b>thus </b><a NAME="E3:37_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">z</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E4:37_1_1_1_1_1_1"><i><font color="Green" title="E17">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c4">y</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Olive" title="b1">v</font>]</span><br/><b>let </b><font color="Maroon" title="c5">v</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c4">y</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">v</font>] )</span><br/><b>assume </b><b>that </b><br/><a NAME="E4:37_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E20">A16</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E5:37_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E21">A17</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c4">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">v</font>]</span><br/><a NAME="E6:37_1_1_1_1_1_1_1_1_2"/>
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 <b>by </b><i><a class="txt" href="fraenkel.html#E4:37_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E20">A16</font></i></a>, <a class="ref" href="setwiseo.html#T6" title="SETWISEO:th.6">SETWISEO:6</a></i>;<br/><b>hence </b><a NAME="E7:37_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">v</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E5:37_1_1_1_1_1_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E19">A15</font></i></a>, <a class="txt" href="fraenkel.html#E5:37_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E21">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:37_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c1">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="Maroon" title="c2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_2"/><i><font color="Green" title="E16">A18</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c1">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="Maroon" title="c2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_1_2_1_1"/>
(  ex <font color="Olive" title="b1">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">w</font>,<font color="Maroon" title="c2">x</font>] ) or for <font color="Olive" title="b1">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> holds <br/> not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">w</font>,<font color="Maroon" title="c2">x</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:37_1_1_1_1_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">w</font>,<font color="Maroon" title="c2">x</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Maroon" title="c3">z</font>] &amp; ( for <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">v</font>,<font color="Olive" title="b1">u</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>] ) )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:37_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E17">A19</font></i>: 
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 <b>and </b><br/><a NAME="E4:37_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E18">A20</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">z</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_2"><i><font color="Green" title="E16">A18</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c5">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E6:37_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E19">A21</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 <b>and </b><br/><a NAME="E7:37_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E20">A22</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">w</font>]
 <b>and </b><br/><a NAME="E8:37_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E21">A23</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">y</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">y</font>,<font color="Olive" title="b1">x</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:37_1_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="fraenkel.html#E3:37_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E17">A19</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c6">u</font> = <font color="Maroon" title="c5">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c3">z</font>] &amp; ( for <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c6">u</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">u</font>,<font color="Olive" title="b1">v</font>] ) )</span><br/><b>thus </b><a NAME="E9:37_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="fraenkel.html#E6:37_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c3">z</font>] &amp; ( for <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c6">u</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">u</font>,<font color="Olive" title="b1">v</font>] ) )</span><br/><b>thus </b><a NAME="E10:37_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c3">z</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E2:37"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="fraenkel.html#E4:37_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="fraenkel.html#E7:37_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E20">A22</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c6">u</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">u</font>,<font color="Olive" title="b1">v</font>]</span><br/><b>let </b><font color="Maroon" title="c7">v</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c6">u</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c7">v</font>] )</span><br/><b>assume </b><b>that </b><br/><a NAME="E11:37_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E22">A24</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E12:37_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E23">A25</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c6">u</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c7">v</font>]</span><br/><a NAME="E13:37_1_1_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> or <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 <b>by </b><i><a class="txt" href="fraenkel.html#E11:37_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E22">A24</font></i></a>, <a class="ref" href="setwiseo.html#T6" title="SETWISEO:th.6">SETWISEO:6</a></i>;<br/><b>hence </b><a NAME="E14:37_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c7">v</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E2:37"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_2"><i><font color="Green" title="E16">A18</font></i></a>, <a class="txt" href="fraenkel.html#E4:37_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="fraenkel.html#E7:37_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="fraenkel.html#E8:37_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E21">A23</font></i></a>, <a class="txt" href="fraenkel.html#E12:37_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E23">A25</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:37_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E17">A26</font></i>: 
for <font color="Olive" title="b1">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> holds <br/> not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">w</font>,<font color="Maroon" title="c2">x</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font>] &amp; ( for <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c2">x</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">v</font>] ) )</span><br/><div class="add"><b>thus </b><a NAME="E2:37_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="setwiseo.html#T6" title="SETWISEO:th.6">SETWISEO:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font>] &amp; ( for <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c2">x</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">v</font>] ) )</span><br/><b>thus </b><a NAME="E3:37_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E18">A27</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:37"><i><font color="Green" title="E10">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c2">x</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">v</font>]</span><br/><b>let </b><font color="Maroon" title="c4">v</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c2">x</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">v</font>] )</span><br/><b>assume </b><b>that </b><br/><a NAME="E4:37_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E19">A28</font></i>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E5:37_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E20">A29</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c2">x</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">v</font>]</span><br/><a NAME="E6:37_1_1_1_1_1_2_1_1_2"/>
not <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E17">A26</font></i></a>, <a class="txt" href="fraenkel.html#E5:37_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E20">A29</font></i></a></i>;<br/><b>hence </b><a NAME="E7:37_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">v</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E3:37_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E18">A27</font></i></a>, <a class="txt" href="fraenkel.html#E4:37_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E19">A28</font></i></a>, <a class="ref" href="setwiseo.html#T6" title="SETWISEO:th.6">SETWISEO:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:37_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c1">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="Maroon" title="c2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:37_1_1_1_1_1_2"><i><font color="Green" title="E16">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:37_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c1">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="Maroon" title="c2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:37_1"/>
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">F<sub>1</sub></font>() holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S4" title="SETWISEO:sch.4">SETWISEO:sch 4</a>(<a class="txt" href="fraenkel.html#E1:37_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="fraenkel.html#E2:37_1"><i><font color="Green" title="E13">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:37_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] &amp; ( for <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">z</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
