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<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="osalg_4.html#V2" title="OSALG_4:attr.2">locally_directed</a> <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">U1</font>, <font color="Olive" title="b2">U2</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">F</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Olive" title="b1">U1</font>,<font color="Olive" title="b2">U2</font>  st <font color="Olive" title="b3">F</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Olive" title="b1">U1</font>,<font color="Olive" title="b2">U2</font> &amp; <font color="Olive" title="b3">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> holds <br/> <a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Olive" title="b1">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Olive" title="b3">F</font></span>)</span>,<font color="Olive" title="b2">U2</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> </span><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">non-empty</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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>  st <font color="Olive" title="b1">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> &amp; <font color="Olive" title="b1">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> holds <br/> <a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Olive" title="b1">F</font></span>)</span>,<font color="Maroon" title="c3">U2</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> </span><br/><b>let </b><font color="Maroon" title="c4">F</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> &amp; <font color="Maroon" title="c4">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> implies  <a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c3">U2</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a>  )</span><br/>





<b>assume </b><a NAME="E1:57"/>
( <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> &amp; <font color="Maroon" title="c4">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c3">U2</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> </span><br/>

<a NAME="E2:57"/><b>then </b>
 <a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a>  <a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c3">U2</font>
 
<b>by </b><i><a class="ref" href="osalg_4.html#T19" target="_self" title="OSALG_4:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E3:57"/>
 <a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c3">U2</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="msualg_3.html#D13" title="MSUALG_3:def.13">MSUALG_3:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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