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<b>reconsider </b><font color="Maroon" title="c1">X</font> = <font color="Maroon">F<sub>2</sub></font>() as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="cqc_sim1.html#E1:35"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>(  not $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or ( $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon">P<sub>1</sub></font>[$1,$2] ) );<br/>
<a NAME="E2:35_1"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c1">X</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> 
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E1:35"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<a NAME="E3:35_1"/><i><font color="Green" title="E25">A5</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] )
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E1:35"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="cqc_sim1.html#E2:35"><i><font color="Green" title="E22">A2</font></i></a></i>;<br/>
<a NAME="E4:35_1"/><i><font color="Green" title="E26">A6</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>]
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E1:35"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="cqc_sim1.html#E3:35"><i><font color="Green" title="E23">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:35_1"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E7:35_1"/><i><font color="Green" title="E28">A8</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font>]
 <b>from </b><i><a class="ref" href="card_3.html#S3" title="CARD_3:sch.3">CARD_3:sch 3</a>(<a class="txt" href="cqc_sim1.html#E2:35_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="cqc_sim1.html#E3:35_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="cqc_sim1.html#E4:35_1"><i><font color="Green" title="E26">A6</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="cqc_sim1.html#E1:35"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="cqc_sim1.html#E6:35_1"><i><font color="Green" title="E27">A7</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <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 <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<b>thus </b><a NAME="E9:35_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E6:35_1"><i><font color="Green" title="E27">A7</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">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 <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<b>let </b><font color="Maroon" title="c4">y</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>] )</span><br/>

<b>assume </b><a NAME="E10:35_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></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="c3">x</font>,<font color="Maroon" title="c4">y</font>]</span><br/>

<b>hence </b><a NAME="E11:35_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>]
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E6:35_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="cqc_sim1.html#E7:35_1"><i><font color="Green" title="E28">A8</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/>


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