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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#V3" title="MSUALG_1:attr.3">strict</a>   <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#R3" title="OSALG_3:pred.3">are_os_isomorphic</a>  holds <br/>( <font color="Olive" title="b1">U1</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>  iff <font color="Olive" title="b2">U2</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">U1</font>, <font color="Maroon" title="c3">U2</font> be   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  implies ( <font color="Maroon" title="c2">U1</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>  iff <font color="Maroon" title="c3">U2</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:26"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">U1</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>  iff <font color="Maroon" title="c3">U2</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>  )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:26"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:26"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c4">F</font> is  <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_3.html#D2" target="_self" title="OSALG_3:def.2">Def2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">O1</font> =  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U1</font>, <font color="Maroon" title="c6">O2</font> =  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">F1</font> = <font color="Maroon" title="c4">F</font> as    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c5">O1</font>,<font color="Maroon" title="c6">O2</font> ;<br/>
<a NAME="E7:26"/>
( <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="msualg_3.html#T13" title="MSUALG_3:th.13">MSUALG_3:13</a></span>;<br/>
<a NAME="E8:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="osalg_3.html#T6" target="_self" title="OSALG_3:th.6">Th7</a></span>;<br/>
<a NAME="E9:26"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a></span>;<br/>
<a NAME="E10:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#D8" title="MSUALG_3:def.8">MSUALG_3:def 8</a></span>;<br/>
<a NAME="E11:26"/><span class="kw">then </span>
 <a href="msualg_3.html#K6" title="MSUALG_3:func.6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="msualg_3.html#T19" title="MSUALG_3:th.19">MSUALG_3:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:26"/>
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="osalg_3.html#T13" target="_self" title="OSALG_3:th.13">Th14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">U2</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>  implies <font color="Maroon" title="c2">U1</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>  )</span><br/>

<span class="kw">reconsider </span><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/>
<span class="kw">assume </span><a NAME="E14:26"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c3">U2</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">U1</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> </span><br/>

<a NAME="E15:26"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="msualg_3.html#T14" title="MSUALG_3:th.14">MSUALG_3:14</a></span>;<br/>
<a NAME="E16:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a></span>;<br/>
<a NAME="E17:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#D8" title="MSUALG_3:def.8">MSUALG_3:def 8</a></span>;<br/>
<a NAME="E18:26"/><span class="kw">then </span>
 <a href="msualg_3.html#K6" title="MSUALG_3:func.6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="msualg_3.html#T19" title="MSUALG_3:th.19">MSUALG_3:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:26"/>
<font color="Maroon" title="c2">U1</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E17:26"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="osalg_3.html#T13" target="_self" title="OSALG_3:th.13">Th14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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