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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> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font><br/> for <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><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> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> &amp; <font color="Olive" title="b3">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b1">U1</font>,<font color="Olive" title="b2">U2</font> holds <br/>(  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Olive" title="b3">F</font> is <a href="osalg_1.html#V12" title="OSALG_1:attr.12">order-sorted</a> &amp;  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Olive" title="b3">F</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font> )</span><br/><b>let </b><font color="Maroon" title="c2">U1</font> be  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</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"> for <font color="Olive" title="b1">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><br/> for <font color="Olive" title="b2">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="Olive" title="b1">U2</font>  st <font color="Olive" title="b2">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> &amp; <font color="Olive" title="b2">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="Olive" title="b1">U2</font> holds <br/>(  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Olive" title="b2">F</font> is <a href="osalg_1.html#V12" title="OSALG_1:attr.12">order-sorted</a> &amp;  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Olive" title="b2">F</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font> )</span><br/><b>let </b><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"> 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> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> &amp; <font color="Olive" title="b1">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> holds <br/>(  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Olive" title="b1">F</font> is <a href="osalg_1.html#V12" title="OSALG_1:attr.12">order-sorted</a> &amp;  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Olive" title="b1">F</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font> )</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> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> &amp; <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> implies (  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c4">F</font> is <a href="osalg_1.html#V12" title="OSALG_1:attr.12">order-sorted</a> &amp;  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c4">F</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font> ) )</span><br/>







<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c4">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> &amp; <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c4">F</font> is <a href="osalg_1.html#V12" title="OSALG_1:attr.12">order-sorted</a> &amp;  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c4">F</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font> )</span><br/>

<b>reconsider </b><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> 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/>
<a NAME="E3:18"/>
<font color="Maroon" title="c4">F</font> <a href="pboole.html#K12" title="PBOOLE:func.12">.:.:</a> <font color="Maroon" title="c5">O1</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="txt" href="#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="osalg_3.html#T8" target="_self" title="OSALG_3:th.8">Th8</a></i>;<br/>
<a NAME="E4:18"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c4">F</font></span>)</span> 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="txt" href="#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="msualg_3.html#D14" title="MSUALG_3:def.14">MSUALG_3:def 14</a></i>;<br/>
<b>hence </b><a NAME="E5:18"/>
 <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c4">F</font> is <a href="osalg_1.html#V12" title="OSALG_1:attr.12">order-sorted</a>
 <b>by </b><i><a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c4">F</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">I</font> =  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c4">F</font> as  <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> <b>by </b><i><a class="txt" href="#E4:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">G</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="c6">I</font><b> such that </b><br/><a NAME="E8:18"/><i><font color="Green" title="E16">A3</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">G</font> &amp; <font color="Maroon" title="c7">G</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="c6">I</font> )
 <b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="msualg_3.html#T21" title="MSUALG_3:th.21">MSUALG_3:21</a></i>;<br/>
<a NAME="E9:18"/><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c7">G</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Maroon" title="c7">G</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> &amp; <font color="Maroon" title="c7">G</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="c6">I</font> )
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E8:18"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a></i>;<br/>
<a NAME="E10:18"/>
for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S1</font>  st <font color="Olive" title="b1">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Olive" title="b2">o2</font> holds <br/> <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o1</font>,<font color="Maroon" title="c6">I</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b2">o2</font>,<font color="Maroon" title="c6">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">o1</font>, <font color="Maroon" title="c9">o2</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</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="c8">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c9">o2</font> implies  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c6">I</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c6">I</font> )</span><br/>

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c8">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c9">o2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c6">I</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c6">I</font></span><br/>

<a NAME="E2:18_1"/><i><font color="Green" title="E19">A6</font></i>: 
(  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c8">o1</font> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c9">o2</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c8">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c9">o2</font> )
 
<b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="osalg_1.html#D22" title="OSALG_1:def.22">OSALG_1:def 22</a></i>;<br/>
<a NAME="E3:18_1"/><i><font color="Green" title="E20">A7</font></i>: 
(  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c6">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c6">I</font> &amp;  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c6">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c6">I</font> )
 
<b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="osalg_1.html#T26" title="OSALG_1:th.26">OSALG_1:26</a></i>;<br/>
<a NAME="E4:18_1"/><i><font color="Green" title="E21">A8</font></i>: 
(  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c2">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c2">U1</font> &amp;  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c2">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c2">U1</font> )
 
<b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="osalg_1.html#T26" title="OSALG_1:th.26">OSALG_1:26</a></i>;<br/>
<a NAME="E5:18_1"/><i><font color="Green" title="E22">A9</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c6">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c6">I</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c6">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c6">I</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:18_1"/><i><font color="Green" title="E23">A10</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="osalg_1.html#D23" title="OSALG_1:def.23">OSALG_1:def 23</a></i>;<br/>
<a NAME="E7:18_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c6">I</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c6">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c10">s1</font> =  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c8">o1</font>;<br/>
<b>set </b><font color="Maroon" title="c11">s2</font> =  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c9">o2</font>;<br/>
<b>let </b><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c6">I</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c6">I</font> )</span><br/>

<b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green" title="E24">A11</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c6">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c6">I</font></span><br/>

<a NAME="E2:18_1_1"/><i><font color="Green" title="E25">A12</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c6">I</font> &amp; <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c6">I</font> )
 
<b>by </b><i><a class="txt" href="#E1:18_1_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c2">U1</font><b> such that </b><br/><a NAME="E4:18_1_1"/><i><font color="Green" title="E26">A13</font></i>: 
<font color="Maroon" title="c7">G</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font>
 <b>by </b><i><a class="txt" href="#E9:18"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="msualg_9.html#T18" title="MSUALG_9:th.18">MSUALG_9:18</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">x</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font>;<br/>
<a NAME="E5:18_1_1"/>
<font color="Maroon" title="c8">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S1</font>
 
;<br/>
<a NAME="E6:18_1_1"/><b>then </b><i><font color="Green" title="E27">A14</font></i>: 
<font color="Maroon" title="c8">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green" title="E28">A15</font></i>:  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c2">U1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">O1</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o1</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D10" title="MSUALG_1:def.10">MSUALG_1:def 10</a></i>
<br/>.= 
<font color="Maroon" title="c5">O1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">o1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:18_1_1"><i><font color="Green" title="E27">A14</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">O1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c8">o1</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D7" title="MSUALG_1:def.7">MSUALG_1:def 7</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c8">o1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E8:18_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c8">o1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c6">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 
<b>by </b><i><a class="txt" href="#E9:18"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="#E4:18_1_1"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="msualg_3.html#D9" title="MSUALG_3:def.9">MSUALG_3:def 9</a></i>;<br/>
<a NAME="E9:18_1_1"/><b>then </b><i><font color="Green" title="E29">A16</font></i>: 
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c8">o1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:18_1_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<a NAME="E10:18_1_1"/><i><font color="Green" title="E30">A17</font></i>: 
( <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c9">o2</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c8">o1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c9">o2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E8:18"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E2:18_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E7:18_1_1"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="osalg_3.html#D1" target="_self" title="OSALG_3:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">y1</font> = <font color="Maroon" title="c14">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c2">U1</font> <b>by </b><i><a class="txt" href="#E4:18_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:18_1_1"/><i><font color="Green" title="E31">A18</font></i>: 
<font color="Maroon" title="c7">G</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c14">y</font>
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E8:18"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E1:18_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="osalg_3.html#T13" target="_self" title="OSALG_3:th.13">Th13</a></i>;<br/>
<a NAME="E13:18_1_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font>
 
<b>by </b><i><a class="txt" href="#E6:18_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E14:18_1_1"/><b>then </b>
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c6">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 
<b>by </b><i><a class="txt" href="#E9:18"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="#E4:18_1_1"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="#E9:18_1_1"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="#E10:18_1_1"><i><font color="Green" title="E30">A17</font></i></a>, <a class="txt" href="#E12:18_1_1"><i><font color="Green" title="E31">A18</font></i></a>, <a class="ref" href="msualg_3.html#D9" title="MSUALG_3:def.9">MSUALG_3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E15:18_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c6">I</font>
 <b>by </b><i><a class="txt" href="#E3:18_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="#E5:18_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="#E2:18_1_1"><i><font color="Green" title="E25">A12</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:18_1"/>
 <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c6">I</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o2</font>,<font color="Maroon" title="c6">I</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:18"/>
 <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c4">F</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font>
 <b>by </b><i><a class="ref" href="osalg_1.html#T27" title="OSALG_1:th.27">OSALG_1:27</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
