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

<b>let </b><font color="Maroon">S</font> be  <a href="osalg_4.html#V2" target="_self">locally_directed</a> <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">U1</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S</font>, <font color="Maroon">U2</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font>;<br/>





<b>assume </b><a NAME="E1:52"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">F</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">F</font> is <a href="osalg_3.html#V1">order-sorted</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">S1</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> as   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="osalg_1.html#T17">OSALG_1:17</a></i>;<br/>
<a NAME="E3:52"/>
 <a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font> is <a href="osalg_4.html#V1" target="_self">os-compatible</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">s2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_4.html#D1" target="_self">OSALG_4:def 1</a><br/>

<b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">s3</font> = <font color="Maroon">s1</font>, <font color="Maroon">s4</font> = <font color="Maroon">s2</font> as   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:52_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> )
 ;<br/>

<a NAME="E4:52_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s4</font>
 
<b>by </b><i>, <a class="ref" href="osalg_1.html#D18">OSALG_1:def 18</a></i>;<br/>
<a NAME="E5:52_1"/><i><font color="Green">E61</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s3</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s3</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s4</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s4</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x</font>, <font color="Maroon">y2</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> <b>by </b><i>, </i>;<br/>
<a NAME="E8:52_1"/><i><font color="Green">E62</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_4.html#K17">MSCng</a> <font color="Maroon">F</font>,<font color="Maroon">s2</font> iff <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y2</font> )
 
<b>by </b><i><a class="ref" href="msualg_4.html#D19">MSUALG_4:def 19</a></i>;<br/>
<a NAME="E9:52_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s4</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s4</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s4</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s4</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> )
 
<b>by </b><i>, , <a class="txt" href="osalg_4.html#E1:52"><i><font color="Green">E33</font></i></a>, <a class="ref" href="osalg_3.html#D1">OSALG_3:def 1</a></i>;<br/>
<a NAME="E10:52_1"/>
( <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K17">MSCng</a> <font color="Maroon">F</font>,<font color="Maroon">s1</font> &amp; <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K17">MSCng</a> <font color="Maroon">F</font>,<font color="Maroon">s2</font> )
 
<b>by </b><i>, <a class="ref" href="msualg_4.html#D20">MSUALG_4:def 20</a></i>;<br/>
<b>hence </b><a NAME="E11:52_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s1</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s2</font> )
 <b>by </b><i>, , <a class="ref" href="msualg_4.html#D19">MSUALG_4:def 19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:52"/>
 <a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font> is   <a href="osalg_4.html#NM4">OSCongruence</a> of <font color="Maroon">U1</font>
 <b>by </b><i/>;<br/>


</div>
