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

<span class="kw">let </span><font color="Maroon" title="c1">X1</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds <br/> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x1</font>] iff <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">x1</font>] ) ) implies <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">y1</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z1</font> where <font color="Olive" title="b1">z1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">z1</font>] </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:53_1"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
for <font color="Olive" title="b1">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds <br/> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x1</font>] iff <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">x1</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">y1</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z1</font> where <font color="Olive" title="b1">z1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">z1</font>] </span>}</span> </span><br/>

<a NAME="E2:53_1"/>
for <font color="Olive" title="b1">X1</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st ( for <font color="Olive" title="b2">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X1</font>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x1</font>] holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">x1</font>] ) holds <br/><span class="p1">{<span class="default"> <font color="Olive" title="b2">y1</font> where <font color="Olive" title="b2">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X1</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">y1</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">z1</font> where <font color="Olive" title="b2">z1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X1</font> : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">z1</font>] </span>}</span> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S5" target="_self" title="DOMAIN_1:sch.5">DOMAIN_1:sch 5</a>();<br/></span>
<span class="kw">hence </span><a NAME="E3:53_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">y1</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z1</font> where <font color="Olive" title="b1">z1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">z1</font>] </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z1</font> where <font color="Olive" title="b1">z1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">z1</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">y1</font>] </span>}</span> </span><br/>

<a NAME="E4:53_1"/>
for <font color="Olive" title="b1">X1</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st ( for <font color="Olive" title="b2">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X1</font>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">x1</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x1</font>] ) holds <br/><span class="p1">{<span class="default"> <font color="Olive" title="b2">y1</font> where <font color="Olive" title="b2">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X1</font> : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">y1</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">z1</font> where <font color="Olive" title="b2">z1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X1</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">z1</font>] </span>}</span> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S5" target="_self" title="DOMAIN_1:sch.5">DOMAIN_1:sch 5</a>();<br/></span>
<span class="kw">hence </span><a NAME="E5:53_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">z1</font> where <font color="Olive" title="b1">z1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">z1</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">y1</font>] </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
