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<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#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S1</font>, <font color="Maroon">U2</font> be  <a href="msualg_1.html#V4">strict</a> <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>assume </b><a NAME="E1:26"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">U1</font>,<font color="Maroon">U2</font> <a href="osalg_3.html#R3" target="_self">are_os_isomorphic</a> 
 ;<br/>

<b>consider </b><font color="Maroon">F</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font><b> such that </b><br/><a NAME="E3:26"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">F</font> <a href="msualg_3.html#R4">is_isomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">F</font> is <a href="osalg_3.html#V1" target="_self">order-sorted</a> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E4:26"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">F</font> is <a href="msualg_3.html#V2">"onto"</a> &amp; <font color="Maroon">F</font> is <a href="msualg_3.html#V1">"1-1"</a> )
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#T13">MSUALG_3:13</a></i>;<br/>
<a NAME="E5:26"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">F</font> <a href="msualg_3.html#K4">""</a>  <a href="msualg_3.html#R4">is_isomorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U1</font>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#T14">MSUALG_3:14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">O1</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>, <font color="Maroon">O2</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</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/>
<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">O1</font>,<font color="Maroon">O2</font> ;<br/>
<a NAME="E8:26"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">F1</font> <a href="msualg_3.html#K4">""</a>  is <a href="osalg_3.html#V1" target="_self">order-sorted</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:26"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">F</font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#D12">MSUALG_3:def 12</a></i>;<br/>
<a NAME="E10:26"/><b>then </b><i><font color="Green">E45</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>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>
<a NAME="E11:26"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="msualg_3.html#K7">Image</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U2</font>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#T19">MSUALG_3:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F2</font> = <font color="Maroon">F1</font> <a href="msualg_3.html#K4">""</a>  as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U2</font>,<font color="Maroon">U1</font> ;<br/>
<a NAME="E13:26"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">F2</font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U1</font>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#D12">MSUALG_3:def 12</a></i>;<br/>
<a NAME="E14:26"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">F2</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U1</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>
<a NAME="E15:26"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="msualg_3.html#K7">Image</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U1</font>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#T19">MSUALG_3:19</a></i>;<br/>
<b>thus </b><a NAME="E16:26"/>
( <font color="Maroon">U1</font> is <a href="osalg_1.html#V13">monotone</a> implies <font color="Maroon">U2</font> is <a href="osalg_1.html#V13">monotone</a> )
 <b>by </b><i>, , , </i>;<br/>

<b>assume </b><a NAME="E17:26"/>
<font color="Maroon">U2</font> is <a href="osalg_1.html#V13">monotone</a>
 ;<br/>

<b>hence </b><a NAME="E18:26"/>
<font color="Maroon">U1</font> is <a href="osalg_1.html#V13">monotone</a>
 <b>by </b><i>, , <a class="ref" href="osalg_3.html#D1" target="_self">Def1</a>, </i>;<br/>


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