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

<b>let </b><font color="Maroon">P1</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span>, <font color="Maroon">P2</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:8_1_2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">x</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> iff  ex <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">T</font> being   <a href="pre_topc.html#L1">TopStruct</a>  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Olive">T</font> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<font color="Olive">V</font></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:8_1_2"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> iff  ex <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">T</font> being   <a href="pre_topc.html#L1">TopStruct</a>  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Olive">T</font> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<font color="Olive">V</font></span>)</span> ) )
 ;<br/>

<a NAME="E3:8_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:8_1_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span> ;<br/>
<a NAME="E3:8_1_2_1"/>
 ex <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">T</font> being   <a href="pre_topc.html#L1">TopStruct</a>  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Olive">T</font> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<font color="Olive">V</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:8_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:8_1_2"/>
<font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:8_1_2_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span> ;<br/>
<a NAME="E3:8_1_2_2"/>
 ex <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">T</font> being   <a href="pre_topc.html#L1">TopStruct</a>  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Olive">T</font> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<font color="Olive">V</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:8_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:8_1_2"/>
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
