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<b>let </b><font color="Maroon" title="c1">S1</font> be   <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon" title="c2">U1</font>, <font color="Maroon" title="c3">U2</font> be  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font>;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> <a href="osalg_3.html#R3" title="OSALG_3:pred.3">are_os_isomorphic</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">F</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font><b> such that </b><br/><a NAME="E3:26"/><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> )
 <b>by </b><i><a class="txt" href="osalg_3.html#E1:26"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="osalg_3.html#D2" target="_self" title="OSALG_3:def.2">Def2</a></i>;<br/>
<a NAME="E4:26"/><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E3:26"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="msualg_3.html#T13" title="MSUALG_3:th.13">MSUALG_3:13</a></i>;<br/>
<a NAME="E5:26"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E3:26"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="msualg_3.html#T14" title="MSUALG_3:th.14">MSUALG_3:14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">O1</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font>, <font color="Maroon" title="c6">O2</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U2</font> as   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font> <b>by </b><i><a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">F1</font> = <font color="Maroon" title="c4">F</font> as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c5">O1</font>,<font color="Maroon" title="c6">O2</font> ;<br/>
<a NAME="E8:26"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c7">F1</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E3:26"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="osalg_3.html#E4:26"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="osalg_3.html#T7" target="_self" title="OSALG_3:th.7">Th7</a></i>;<br/>
<a NAME="E9:26"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E3:26"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="msualg_3.html#D12" title="MSUALG_3:def.12">MSUALG_3:def 12</a></i>;<br/>
<a NAME="E10:26"/><b>then </b><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a></i>;<br/>
<a NAME="E11:26"/><b>then </b><i><font color="Green" title="E23">A8</font></i>: 
 <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">U2</font>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E9:26"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="msualg_3.html#T19" title="MSUALG_3:th.19">MSUALG_3:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">F2</font> = <font color="Maroon" title="c7">F1</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c2">U1</font> ;<br/>
<a NAME="E13:26"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c8">F2</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E5:26"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="msualg_3.html#D12" title="MSUALG_3:def.12">MSUALG_3:def 12</a></i>;<br/>
<a NAME="E14:26"/><b>then </b><i><font color="Green" title="E25">A10</font></i>: 
<font color="Maroon" title="c8">F2</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a></i>;<br/>
<a NAME="E15:26"/><b>then </b><i><font color="Green" title="E26">A11</font></i>: 
 <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c8">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E13:26"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="msualg_3.html#T19" title="MSUALG_3:th.19">MSUALG_3:19</a></i>;<br/>
<b>thus </b><a NAME="E16:26"/>
( <font color="Maroon" title="c2">U1</font> is <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> implies <font color="Maroon" title="c3">U2</font> is <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> )
 <b>by </b><i><a class="txt" href="osalg_3.html#E3:26"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="osalg_3.html#E10:26"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="osalg_3.html#E11:26"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="osalg_3.html#T14" target="_self" title="OSALG_3:th.14">Th14</a></i>;<br/>

<b>assume </b><a NAME="E17:26"/>
<font color="Maroon" title="c3">U2</font> is <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>
 ;<br/>

<b>hence </b><a NAME="E18:26"/>
<font color="Maroon" title="c2">U1</font> is <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>
 <b>by </b><i><a class="txt" href="osalg_3.html#E8:26"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="osalg_3.html#E14:26"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="osalg_3.html#E15:26"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="osalg_3.html#T14" target="_self" title="OSALG_3:th.14">Th14</a></i>;<br/>


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