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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <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>3</sub></font>() st <br/>( <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">y</font>] &amp; ( for <font color="Olive" title="b2">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 iff  ex <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">x</font>] ) ) ) );<br/>
<a NAME="E1:53_1"/><i><font color="Green" title="E23">A1</font></i>: 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">A</font>,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span> ,<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span> </span>)</span>,<font color="Maroon">F<sub>1</sub></font>()
 
<b>from </b><i><a class="ref" href="yellow_3.html#S5" title="YELLOW_3:sch.5">YELLOW_3:sch 5</a>();<br/></i>
<a NAME="E2:53_1"/><i><font color="Green" title="E24">A2</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">y</font>] ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_1_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">y</font>] ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">y</font>] ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span>  or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">y</font>] ) </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">y</font>] ) </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c2">A1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:53_1_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">A2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E5:53_1_1_1"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">A2</font>] )
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E3:53_1_1_1"><i><font color="Green" title="E25">A3</font></i></a></i>;<br/>
<b>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>3</sub></font>()<b> such that </b><br/><a NAME="E7:53_1_1_1"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">y</font>]
 <b>and </b><br/><a NAME="E8:53_1_1_1"/><i><font color="Green" title="E28">A6</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A2</font> iff  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Maroon" title="c4">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] ) )
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E5:53_1_1_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<a NAME="E9:53_1_1_1"/>
 ex <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>2</sub></font>() st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E3:53_1_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="lfuzzy_0.html#E5:53_1_1_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="lfuzzy_0.html#E8:53_1_1_1"><i><font color="Green" title="E28">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:53_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">y</font>] ) </span>}</span> 
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E7:53_1_1_1"><i><font color="Green" title="E27">A5</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>thus </b><a NAME="E2:53_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">y</font>] ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</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="53_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">y</font>] ) </span>}</span>  or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">y</font>] ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Maroon" title="c3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E3:53_1_1_2"/><i><font color="Green" title="E25">A7</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">y</font>] )
 ;<br/>
<b>set </b><font color="Maroon" title="c4">A1</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x'</font>,<font color="Maroon" title="c3">y</font>) where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x'</font>] </span>}</span> ;<br/>
<a NAME="E4:53_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x'</font>,<font color="Maroon" title="c3">y</font>) where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x'</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x'</font>,<font color="Maroon" title="c3">y</font>) where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x'</font>] </span>}</span>  or <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_2_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x'</font>,<font color="Maroon" title="c3">y</font>) where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x'</font>] </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><br/>

<a NAME="E2:53_1_1_2_1"/><b>then </b>
 ex <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>2</sub></font>() st <br/>( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x'</font>,<font color="Maroon" title="c3">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x'</font>] )
 
;<br/>
<b>hence </b><a NAME="E3:53_1_1_2_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">A1</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x'</font>,<font color="Maroon" title="c3">y</font>) where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x'</font>] </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E6:53_1_1_2"/><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E3:53_1_1_2"><i><font color="Green" title="E25">A7</font></i></a></i>;<br/>
<a NAME="E7:53_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">A1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_2_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c3">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>2</sub></font>[<font color="Maroon" title="c3">y</font>] &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> iff  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Maroon" title="c3">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] ) ) ) )</span><br/>

<b>thus </b><a NAME="E1:53_1_1_2_2"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">y</font>]
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E3:53_1_1_2"><i><font color="Green" title="E25">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">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> iff  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Maroon" title="c3">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] ) )</span><br/>

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> iff  ex <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>2</sub></font>() st <br/>( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )</span><br/>

<b>thus </b><a NAME="E2:53_1_1_2_2"/>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> iff  ex <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>2</sub></font>() st <br/>( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 ; <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>
<a NAME="E8:53_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E9:53_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E6:53_1_1_2"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>
<a NAME="E3:53_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">A</font>,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] </span>}</span> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">A</font>,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] </span>}</span> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] </span>}</span> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">A</font>,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">A</font>,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span>  or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] </span>}</span> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:53_1_2_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">A</font>,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] </span>}</span> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c2">A1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:53_1_2_1"/><i><font color="Green" title="E25">A9</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c2">A1</font>,<font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A1</font>] )
 ;<br/>
<b>consider </b><font color="Maroon" title="c3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E5:53_1_2_1"/><i><font color="Green" title="E26">A10</font></i>: 
( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">y</font>] &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A1</font> iff  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Maroon" title="c3">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] ) ) ) )
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E3:53_1_2_1"><i><font color="Green" title="E25">A9</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A1</font> iff <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  )</span><br/><a NAME="E1:53_1_2_1_1"/>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  iff  ex <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>2</sub></font>() st <br/>( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 ;<br/><b>hence </b><a NAME="E2:53_1_2_1_1"/>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A1</font> iff <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  )
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E5:53_1_2_1"><i><font color="Green" title="E26">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>
<a NAME="E7:53_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E8:53_1_2_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] </span>}</span> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E3:53_1_2_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="lfuzzy_0.html#E5:53_1_2_1"><i><font color="Green" title="E26">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>thus </b><a NAME="E2:53_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] </span>}</span> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">A</font>,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</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="53_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] </span>}</span> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span>  or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">A</font>,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:53_1_2_2"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] </span>}</span> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">A</font>,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E3:53_1_2_2"/><i><font color="Green" title="E25">A11</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> ,<font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c2">y</font>] )
 ;<br/>
<a NAME="E4:53_1_2_2"/>
 ex <font color="Olive" title="b1">A1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">A1</font>,<font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A1</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c3">A1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c4">A2</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> ;<br/>
<a NAME="E2:53_1_2_2_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  or <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() )</span><br/>

<b>assume </b><a NAME="E1:53_1_2_2_1_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><br/>

<a NAME="E2:53_1_2_2_1_1"/><b>then </b>
 ex <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>2</sub></font>() st <br/>( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 
;<br/>
<b>hence </b><a NAME="E3:53_1_2_2_1_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">A1</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E4:53_1_2_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">A1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_2_1_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c2">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>2</sub></font>[<font color="Maroon" title="c2">y</font>] &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> iff  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Maroon" title="c2">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] ) ) ) )</span><br/>

<b>thus </b><a NAME="E1:53_1_2_2_1_2"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c2">y</font>]
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E3:53_1_2_2"><i><font color="Green" title="E25">A11</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">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> iff  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Maroon" title="c2">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] ) )</span><br/>

<b>thus </b><a NAME="E2:53_1_2_2_1_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> iff  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Maroon" title="c2">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] ) )
 ; <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="E5:53_1_2_2_1"/>
 ex <font color="Olive" title="b1">A1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">A1</font>,<font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A1</font>] )
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E3:53_1_2_2"><i><font color="Green" title="E25">A11</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="E5:53_1_2_2"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">A</font>,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span>}</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>
<b>hence </b><a NAME="E4:53_1"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] </span>}</span> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span> ,<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">y</font>] ) </span>}</span> ,<font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E1:53_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="lfuzzy_0.html#E2:53_1"><i><font color="Green" title="E24">A2</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>
