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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>; <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">S1</font>  st <font color="Olive" title="b1">U1</font>,<font color="Olive" title="b2">U2</font> <a href="osalg_3.html#R1" title="OSALG_3:pred.1">are_os_isomorphic</a>  holds <br/><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b1">U1</font> <a href="osalg_3.html#R1" title="OSALG_3:pred.1">are_os_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">S1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> <a href="osalg_3.html#R1" title="OSALG_3:pred.1">are_os_isomorphic</a>  implies <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c2">U1</font> <a href="osalg_3.html#R1" title="OSALG_3:pred.1">are_os_isomorphic</a>  )</span><br/>



<b>assume </b><a NAME="E1:12"/>
<font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> <a href="osalg_3.html#R1" title="OSALG_3:pred.1">are_os_isomorphic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c2">U1</font> <a href="osalg_3.html#R1" title="OSALG_3:pred.1">are_os_isomorphic</a> </span><br/>

<b>then 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="E3:12"/><i><font color="Green" title="E11">A1</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="ref" href="osalg_3.html#D2" target="_self" title="OSALG_3:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">G</font> = <font color="Maroon" title="c4">F</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="E5:12"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c5">G</font> <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="#E3:12"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="msualg_3.html#T14" title="MSUALG_3:th.14">MSUALG_3:14</a></i>;<br/>
<a NAME="E6:12"/><i><font color="Green" title="E13">A3</font></i>: 
( the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U1</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font> &amp; the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">U2</font> is   <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/>
<a NAME="E7:12"/>
( <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="#E3:12"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="msualg_3.html#T13" title="MSUALG_3:th.13">MSUALG_3:13</a></i>;<br/>
<a NAME="E8:12"/><b>then </b>
<font color="Maroon" title="c4">F</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="#E3:12"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E6:12"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="osalg_3.html#T7" target="_self" title="OSALG_3:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E9:12"/>
<font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c2">U1</font> <a href="osalg_3.html#R1" title="OSALG_3:pred.1">are_os_isomorphic</a> 
 <b>by </b><i><a class="txt" href="#E5:12"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="osalg_3.html#D2" target="_self" title="OSALG_3:def.2">Def2</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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