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

<b>let </b><font color="Maroon">J</font> be  <a href="yellow_1.html#V1" target="_self">RelStr-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of  <a href="xboole_0.html#K1">{}</a> ;<br/>

<b>set </b><font color="Maroon">IT</font> =  <a href="yellow_1.html#K5" target="_self">product</a> <font color="Maroon">J</font>;<br/>
<i><font color="Green">E21</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <font color="Maroon">J</font></span>)</span> = 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="card_3.html#T19">CARD_3:19</a>, <a class="ref" href="pboole.html#T134">PBOOLE:134</a></i>
;<br/>
<a NAME="E2:48"/><i><font color="Green">E24</font></i>: 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="card_3.html#T19">CARD_3:19</a>, <a class="ref" href="pboole.html#T134">PBOOLE:134</a></i>;<br/>
<a NAME="E3:48"/>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a><br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:48_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <font color="Maroon">J</font></span>)</span>
 ;<br/><a NAME="E2:48_2_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <font color="Maroon">J</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <font color="Maroon">J</font></span>)</span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:48_2_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:48_2_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:48_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:48_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:48_2"/><b>then </b><i><font color="Green">E25</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E4:48_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> =  <a href="xboole_0.html#K1">{}</a> , <font color="Maroon">y</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <font color="Maroon">J</font></span>)</span> <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:48_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="pre_circ.html#K2">--&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="pre_circ.html#K2">--&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font>, <font color="Maroon">g</font> = <font color="Maroon">y</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E8:48_2"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</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">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> )
 
;<br/>
<a NAME="E9:48_2"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E10:48_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <font color="Maroon">J</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:48"/>
 <a href="yellow_1.html#K5" target="_self">product</a> <font color="Maroon">J</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> #)
 <b>by </b><i/>;<br/>


</div>
