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

<b>let </b><font color="Maroon">S1</font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">OU0</font> be   <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">OU0</font>;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font>, <font color="Maroon">s2</font> be   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font>;<br/>







<b>assume </b><a NAME="E1:41"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s2</font>
 ;<br/>

<a NAME="E2:41"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K8" target="_self">OSSubSort</a> <font color="Maroon">A</font>,<font color="Maroon">s2</font> holds <br/> ex <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K8" target="_self">OSSubSort</a> <font color="Maroon">A</font>,<font color="Maroon">s1</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K8" target="_self">OSSubSort</a> <font color="Maroon">A</font>,<font color="Maroon">s2</font>
 ;<br/>

<b>consider </b><font color="Maroon">B</font> being    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">OU0</font><b> such that </b><br/><a NAME="E3:41_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K5" target="_self">OSSubSort</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> )
 <b>by </b><i><a class="txt" href="osalg_2.html#E1:41"><i><font color="Green">E25</font></i></a>, </i>;<br/>
<a NAME="E4:41_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">B</font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S1</font>
 
<b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
;<br/>

<b>thus </b><a NAME="E5:41_1"/>
( <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K8" target="_self">OSSubSort</a> <font color="Maroon">A</font>,<font color="Maroon">s1</font> &amp; <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="txt" href="osalg_2.html#E1:41_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="osalg_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="osalg_1.html#D18">OSALG_1:def 18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:41"/>
 <a href="osalg_2.html#K8" target="_self">OSSubSort</a> <font color="Maroon">A</font>,<font color="Maroon">s2</font> <a href="setfam_1.html#R2">is_coarser_than</a>  <a href="osalg_2.html#K8" target="_self">OSSubSort</a> <font color="Maroon">A</font>,<font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D3">SETFAM_1:def 3</a></i>;<br/>


</div>
