<?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">U1</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#V13">monotone</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S1</font>;<br/><b>let </b><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">S1</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="E1:18"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">F</font> is <a href="osalg_3.html#V1" target="_self">order-sorted</a> &amp; <font color="Maroon">F</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">O1</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">S1</font> <b>by </b><i><a class="ref" href="osalg_1.html#T17">OSALG_1:17</a></i>;<br/>
<a NAME="E3:18"/>
<font color="Maroon">F</font> <a href="pboole.html#K13">.:.:</a> <font color="Maroon">O1</font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:18"/><b>then </b><i><font color="Green">E32</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">F</font></span>)</span> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S1</font>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#D14">MSUALG_3:def 14</a></i>;<br/>
<b>hence </b><a NAME="E5:18"/>
 <a href="msualg_3.html#K7">Image</a> <font color="Maroon">F</font> is <a href="osalg_1.html#V12">order-sorted</a>
 <b>by </b><i><a class="ref" href="osalg_1.html#T17">OSALG_1:17</a></i>;<br/>

<b>reconsider </b><font color="Maroon">I</font> =  <a href="msualg_3.html#K7">Image</a> <font color="Maroon">F</font> as  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S1</font> <b>by </b><i>, <a class="ref" href="osalg_1.html#T17">OSALG_1:17</a></i>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">I</font><b> such that </b><br/><a NAME="E8:18"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">F</font> <a href="pboole.html#R6">=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">I</font> )
 <b>by </b><i>, <a class="ref" href="msualg_3.html#T21">MSUALG_3:21</a></i>;<br/>
<a NAME="E9:18"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">G</font> is <a href="msualg_3.html#V2">"onto"</a> &amp; <font color="Maroon">G</font> is <a href="osalg_3.html#V1" target="_self">order-sorted</a> &amp; <font color="Maroon">G</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">I</font> )
 
<b>by </b><i>, , <a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>
<a NAME="E10:18"/>
for <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S1</font>  st <font color="Olive">o1</font> <a href="osalg_1.html#R6">&lt;=</a> <font color="Olive">o2</font> holds <br/> <a href="msualg_1.html#K5">Den</a> <font color="Olive">o1</font>,<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Olive">o2</font>,<font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o1</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S1</font>, <font color="Maroon">o2</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S1</font>;<br/>

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">o1</font> <a href="osalg_1.html#R6">&lt;=</a> <font color="Maroon">o2</font>
 ;<br/>

<a NAME="E2:18_1"/><i><font color="Green">E44</font></i>: 
(  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font> <a href="osalg_1.html#R2">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o2</font> )
 
<b>by </b><i>, <a class="ref" href="osalg_1.html#D22">OSALG_1:def 22</a></i>;<br/>
<a NAME="E3:18_1"/><i><font color="Green">E45</font></i>: 
(  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o2</font>,<font color="Maroon">I</font> &amp;  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o1</font>,<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o2</font>,<font color="Maroon">I</font> )
 
<b>by </b><i>, <a class="ref" href="osalg_1.html#T26">OSALG_1:26</a></i>;<br/>
<a NAME="E4:18_1"/><i><font color="Green">E50</font></i>: 
(  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o2</font>,<font color="Maroon">U1</font> &amp;  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o2</font>,<font color="Maroon">U1</font> )
 
<b>by </b><i>, <a class="ref" href="osalg_1.html#T26">OSALG_1:26</a></i>;<br/>
<a NAME="E5:18_1"/><i><font color="Green">E51</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">I</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">I</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">I</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o2</font>,<font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:18_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">U1</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font>
 
<b>by </b><i>, <a class="ref" href="osalg_1.html#D23">OSALG_1:def 23</a></i>;<br/>
<a NAME="E7:18_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">I</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">s1</font> =  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font>;<br/>
<b>set </b><font color="Maroon">s2</font> =  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o2</font>;<br/>
<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> ;<br/>

<b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green">E53</font></i>: 
<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="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:18_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">I</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o1</font>,<font color="Maroon">I</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font><b> such that </b><br/><a NAME="E4:18_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">G</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="msualg_9.html#T18">MSUALG_9:18</a></i>;<br/>
<b>set </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>;<br/>
<a NAME="E5:18_1_1"/>
<font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font>
 
;<br/>
<a NAME="E6:18_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">o1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E62</font></i>:  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">O1</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>
<br/>.= 
<font color="Maroon">O1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">O1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E8:18_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="osalg_3.html#D1" target="_self">Def1</a>, <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>;<br/>
<a NAME="E9:18_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E10:18_1_1"/><i><font color="Green">E88</font></i>: 
( <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</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">G</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o2</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> )
 
<b>by </b><i>, , , , </i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o2</font>,<font color="Maroon">U1</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:18_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">G</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E13:18_1_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E14:18_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="osalg_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="osalg_3.html#T2" target="_self">Th2</a>, <a class="txt" href="osalg_3.html#E1:18"><i><font color="Green">E31</font></i></a>, <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E15:18_1_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="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">I</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:18_1"/>
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:18"/>
 <a href="msualg_3.html#K7">Image</a> <font color="Maroon">F</font> is  <a href="osalg_1.html#V13">monotone</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="osalg_1.html#T27">OSALG_1:27</a></i>;<br/>


</div>
