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

<b>thus </b><a NAME="E1:23_1_2"/>
for <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2">EqRelLatt</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> holds <br/> ( <font color="Olive">Y</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font> iff for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> iff  ex <font color="Olive">Eq</font> being  <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">Eq</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">Eq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2">EqRelLatt</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>;<br/>

<b>thus </b><a NAME="E1:23_1_2_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font> implies for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff  ex <font color="Olive">Eq</font> being  <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">Eq</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">Eq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_2_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E2:23_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> implies  ex <font color="Olive">Eq</font> being  <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">Eq</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">Eq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:23_1_2_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i>, <a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E5:23_1_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:23_1_2_1_1"/>
(  ex <font color="Olive">Eq</font> being  <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">Eq</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">Eq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:23_1_2_1"/>
( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff  ex <font color="Olive">Eq</font> being  <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">Eq</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">Eq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) ) ) implies <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_2_1_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff  ex <font color="Olive">Eq</font> being  <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">Eq</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">Eq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 ;<br/>

<b>thus </b><a NAME="E2:23_1_2_1_2"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_2_1_2_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Eq</font> being   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:23_1_2_1_2_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Eq</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">Eq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:23_1_2_1_2_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:23_1_2_1_2"/>
 <a href="card_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_2_1_2_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:23_1_2_1_2_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i>, <a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/>
<a NAME="E5:23_1_2_1_2_2"/>
 ex <font color="Olive">Eq</font> being  <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> st <br/>( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">Eq</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">Eq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1_2_2_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E1:23_1_2_1_2_2_1"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:23_1_2_1_2_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>


</div><b>end;</b></div>


</div>
