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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font></span>)</span> where <font color="Olive" title="b1">s1</font> is   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> : <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> $1 </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:18_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font> holds  <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">d</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S4" title="FUNCT_1:sch.4">FUNCT_1:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">f</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> <b>by </b><i><a class="txt" href="osalg_2.html#E2:18_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">f1</font> = <font color="Maroon" title="c4">f</font> as   <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <font color="Maroon" title="c1">S1</font> ;<br/>
<a NAME="E5:18_1_1"/>
<font color="Maroon" title="c5">f1</font> is <a href="osalg_1.html#V11" title="OSALG_1:attr.11">order-sorted</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">r1</font>, <font color="Maroon" title="c7">r2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D18" title="OSALG_1:def.18">OSALG_1:def 18</a><br/>

<b>assume </b><a NAME="E1:18_1_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c6">r1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">r2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:18_1_1_1"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">r1</font>
 ;<br/>

<a NAME="E3:18_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s2</font></span>)</span> where <font color="Olive" title="b1">s2</font> is   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> : <font color="Olive" title="b1">s2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">r1</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="osalg_2.html#E2:18_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="osalg_2.html#E2:18_1_1_1"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:18_1_1_1"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s2</font></span>)</span> where <font color="Olive" title="b1">s2</font> is   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> : <font color="Olive" title="b1">s2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">r1</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">s3</font> being   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font><b> such that </b><br/><a NAME="E7:18_1_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s3</font> &amp; <font color="Maroon" title="c10">s3</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">r1</font> )
 <b>by </b><i><a class="txt" href="osalg_2.html#E5:18_1_1_1"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/>
<a NAME="E8:18_1_1_1"/>
<font color="Maroon" title="c10">s3</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">r2</font>
 
<b>by </b><i><a class="txt" href="osalg_2.html#E1:18_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="osalg_2.html#E7:18_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>
<a NAME="E9:18_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s2</font></span>)</span> where <font color="Olive" title="b1">s2</font> is   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> : <font color="Olive" title="b1">s2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">r2</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="osalg_2.html#E7:18_1_1_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<a NAME="E10:18_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s2</font></span>)</span> where <font color="Olive" title="b1">s2</font> is   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> : <font color="Olive" title="b1">s2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">r2</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="osalg_2.html#E5:18_1_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:18_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r2</font>
 <b>by </b><i><a class="txt" href="osalg_2.html#E2:18_1_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">f2</font> = <font color="Maroon" title="c5">f1</font> as   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">f2</font>
;<br/>

<b>thus </b><a NAME="E7:18_1_1"/>
for <font color="Olive" title="b1">s</font> being  <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> holds  <font color="Maroon" title="c6">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b2">s1</font> is   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> : <font color="Olive" title="b2">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="osalg_2.html#E2:18_1_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>


</div>
